Which Pokémon "types" are the most well-rounded?
Solution 1:
It's not about the individual
As was said in Williham Totland's answer, it's about the overall typing of the team, not the typing of the individual pokemon. No one pokemon will be able to rise to any challenge you face, but that is why you have room to carry six with you.
A good starting point is to pick pokemon with different typings, both offensively and defensively. Let's look at an example of what I mean by this.
Example Pokemon
Defensive Analysis
Blastoise is a Water pokemon, so he's going to be strong against fire, ice, steel, and water attacks, but weak to electric and grass attacks. It may look like Blastoise is a good choice because he's strong against more things than he's weak against, but if you look more closely, it actually comes out pretty evenly. Electric and Grass are two very prevalent types of moves, especially grass. You can expect almost every trainer and grass patch in the first half of the game to have a grass move of some kind. Conversely, being strong against steel isn't that big a deal because those moves are very rare and the pokemon who use them generally don't have a very strong attack power.
Offensive Analysis
Offensively, Blastoise gives access to STAB surf, which is extremely strong, as well as some ice attacks if you use a TM. He really doesn't have too much versatility at all.
Example Team
Those are the kinds of things you should take into account when looking at an individual Pokemon, but what about a team? Let's say you have a Venusaur, Golem, Pidgeot, and Dragonite. There are a lot of different types represented here, so at first glance it may look like it's well-rounded. The problem, however, is the glaring weakness to Ice. Venusaur, Golem, Pidgeot, and Dragonite are all weak to ice. If you don't do something about this, you're probably going to have trouble taking on any kind of ice trainers once you get later on. You may want to consider adding in Rapidash or Primeape.
General Strategy
The general principle here is to take a look at your team as a whole. You want to try to use pokemon who have a wide array of offensive options and few defensive drawbacks. You also want to make sure that, out of your six, you have a few ways to deal with common threats and at least one way to deal with any kind of threat. It's okay if you don't have a fire attack on your team, as long as you have a flying attack and a fighting attack to cover the cases where fire would have been usefel. (Fire is strong against grass, ice, and bug, so having flying (strong against grass/bug) and fighting (strong against ice) will cover the loss of a fire move.)
Generally Good Types
Particularly strong offensive types for playing through casually are:
- flying
- fighting
- ground
- rock
Particularly strong defensive types for playing through casually are:
- normal
- rock
- steel
(Note that all of these defensive types are weak to fighting, so make sure to include something to compensate for that.)
One Last Bit
The best advice I can give, to wrap it all up, is to pay attention to what you have difficulty with, and adjust to compensate for that. If you are consistently having trouble taking down electric pokemon, add a ground pokemon to your lineup. If your Jigglypuff never contributes anything positive to the fight and just comes in to give you a turn to heal up someone else, maybe consider replacing it with a better alternative.
Sorry for using all first gen Pokemon in my examples. I don't know fifth gen as well off the top of my head, and I think this may make the answer more accessible to more people anyway
Solution 2:
Essentially, you don't really need to care about the "roundedness" of individual Pokémon; what you need to consider is how versatile your team is.
More than that, types aren't as important as roles: You're going to want to build a balanced team that can deal with most types, but certain types usually fill certain roles: You rarely see a defensive Fire-type; and you rarely see an offensive Steel-type.
All types are, by design, unbalanced. What matters is the team.
Edit: Also, the most "balanced" and "rounded" type is likely the Normal type.
Solution 3:
The term "Well-Rounded" has so many factors that both casual and competitive players keep in mind when choosing their Pokemon, but most importantly, picking out THEIR TEAM.
First off:
Casual Gamers
Weaknesses:
If you're considering a "Well-Rounded" approach, consider using Pokemon that have few or no notable weaknesses. Pokemon like Spiritomb, or Eelektross* are among a few Pokemon that have no known weaknesses (meaning, no super-effective attacks on them).
As other people have mentioned above, try picking Pokemon that are multi-typed, meaning, they carry more than one "type". Having more than one type usually opens up more movesets for your Pokemon, allowing you to fit a lot more different and powerful moves into your team. For example, having a Thunder/Water Pokemon is nice because, although Pokemon with only the Water type would take super-effective damage from Thunder-types, a multi-type does not face such problems, and helps to cover any gaps in a Pokemon's abilities.
Ability types:
You'll also want to consider different "ability" types. Certain Pokemon have better ability types than others, some good, some not so good. For example, the pokemon Shedinja has the ability Wonder Guard, which nullifies all damage that isn't a super-effective attack. Note that Shedinja can only have a maximum of 1 hp, but that means the only attacks that can actually kill it are Dark, Fire, Flying, Ghost, and Rock, which means you have to only worry about Pokemon with those type attacks.
Another example is Yanma, which one of his abilities, Compound Eyes, grants a 1.3x bonus to accuracy. This can help fill in some moves in the game, for example, if you have a move that doesn't have a 100 accuracy (meaning it may miss), Compound Eyes increases it to the point where it's almost a guaranteed hit. So consider having different Pokemon with different abilities; there are a TON out there to choose from.
Now since you're playing casual, and not getting into the messy world of Competitive players, you might want to consider getting an HM mule. Since most casual players spend their time travelling throughout Unova exploring places, it might be a good idea to carry Pokemon with abilities that can be used both on and off the map.
Mules:
A good example is Bibarrel, a more famous pack mule. Because HM's are so hard to get rid of (once you progress to the end of the game, there should be a HM remover trainer), and usually not that strong, it may be more effective to give HMs to another Pokemon in that party to take care of. As I mentioned, Bibarrel is capable of taking the HMs Cut, Surf, Strength, Waterfall, Dive. which helps to cover a lot of the travelling you'll do throughout Unova. Another example is Pelipper, which can learn the mvoes Fly and Surf, two of the most important moves in game.
Variety
You'll definitely want to have a multitude of different Pokemon in your team. Say for example you go up against a Fire type Pokemon, and unfortunately, your whole team is full of Sunkern. Obviously, you're going to be in for a huge thrashing, that Fire type will raze through your whole team. Now say we had the same situation, but you had 5 sunkerns (not much better), and 1 water type. This means you actually stand a chance, and can possible take down that Fire type pokemon. Basically, don't just center your team around one type of Pokemon, or one type, try mixing it up, and trying to vary it a bit. Having extra Pokemon that fit certain roles is always good; it's pretty much a good way to switch up your teams.
STAB
STAB stands for Same-Type Attack Bonus. Basically, let's say you're a water Pokemon, and you have a water type move, Surf. Surf has a base power of 95. What STAB basically does is, if your Pokemon has the same type as the type of the attack, it adds 50% bonus damage to your attack. Meaning, you should match moves of the same type as your Pokemon, and the same for multi-type Pokemon. As mentioned before, the Pokemon Pelliper has a typing of Water/Flying. If you teach it Fly, and Surf, that means you get a damage bonus whenever you use Fly, because it's a STAB bonus, and the same for SURF. Keep this in mind to make the most out of your attacks.
TL;DR Casual Version Pick Pokemon with multiple types; this will allow a broad range of moves and different abilities. Having different Pokemon in your team, as well as in your PC, works wonders, meaning you can substitute what gaps you need to fill any time.
Competitive Gamers
I'm not much of a Competitive player; the scene there is a lot of experienced, well-polished players who could probably take me down any time. In a competitive setting, there is a bigger emphasis on the team aspect, meaning, all the aspects of the team should synergize well enough with another.
Teams: When playing competitively, there is a huge emphasis on crafting the right team with the right Pokemon with the right EVs and IVs (i'll get to it in a bit), with the right abilities, with the right moves, all with the right items along with the right nature. Yeah, that's quite a mouthful. Basically, every single detail you probably wouldn't care about in a Casual perspective is taken in, magnified to the maximum detail, and carefully analyzed to produce the best team. Since there are fairly notable Pokemon to watch out in competitive matches, meaning, Pokemon that are fairly commonplace on teams because they are THAT good.
Movesets: This is where Competitive players may consider using moves which have high attack levels (usually at the cost of nerfing some stats), priority moves (think Quick Attack), or even Buff Moves (think Defend (raises Defense)). Basically, it goes down to a whole huge world of detail I'm not too experienced with.
EVs and IVs: I'm not too good with this, I'll just leave a really comprehensive link here, as I don't want to take their credit. Basically, the jist of the matter is, when you defeat certain Pokemon, you end up gaining a certain amount of "Effort Value (EV)" points that contribute towards certain statistics.
TL;DR Competitive Version It takes a lot of time planning, making teams, setting up teams, and considering each battle turn by turn to get really good at this.
Other Notable PokeLinks: Some good websites to frequent if you ever get into Competitive gaming, or even Casual gaming are Smogon (competitive), Veekun (virtual Pokedex), Serebii, Bulbapedia.
*Though Eelektross does have a weakness to Ground based attacks, his ability Levitate completely nullifies incoming Ground type damage, meaning this Pokemon has no virtual weakness.