How can I kill puppies without consequences?

As you guessed, killing your pet either directly or via some indirect actions (such as displacing it into a body of water) can potentially anger your god or affect your stats. However, yes, there are a few ways that you can separate yourself from your pet, with no negative consequences:

• Abandon it by changing floors while it's more than one square away from you.
• Let it get killed in combat with a tough dungeon monster. The Gnomish Mines are a pretty good early-game place to find such monsters.
• Let it get caught in a trap (often easier to do by accident than on purpose)
• Dip or quaff from a fountain, and let it get killed by the monsters that may appear
• Use a scroll of genocide (unlikely and incredibly wasteful)

All that said, you may want to reconsider. If you let them get enough XP to level up properly, even your starting pets can become incredibly powerful, and will be very helpful to you in combat during the earlier parts of the game. If you're getting annoyed with your pet's movement patterns, using a leash (if you can find one) will keep your pet closer to you, and keep it from moving quite so erratically. Additionally, later into the game there are some incredibly powerful dungeon monsters that you can tame to be your pet as well, which can be a huge benefit.

Yes, there's a penalty of −15 alignment and −1 luck for killing your pet.

The alignment penalty is actually pretty insignificant, since it's easy to make up for by just killing a few hostile monsters, but the luck penalty can hurt you (mainly by decreasing your odds of hitting anything, as well as in several direct ways), especially if you get several luck penalties at the same time. (−1 luck is survivable for a while; −6 luck just plain sucks.) It does, however, eventually wear out in about 600 turns, unless you happen to be carrying a non-blessed luckstone.

Also note that, no matter how your pet died, sacrificing its corpse is a bad idea. (Sacrificing corpses of ex-pets that went untame before they died is OK, though.) Eating a former pet is actually not a problem as such, but you shouldn't eat cats or dogs unless you're a caveman or an orc.

If you don't want to start the game with a pet following you around, the absolutely easiest and most reliable way is to add the line:

OPTIONS=pettype:none

to your nethack config file (.nethackrc / defaults.nh). That way, your character will never start with a pet. (Yes, this option will also make knights start without a horse — this isn't generally much of a loss, except for the fact that horseless knights will also lose the free saddle it comes with.)

If you did start with a pet and want to get rid of it, the second-easiest option is to simply leave your pet behind when you move to a new level. Somewhat curiously, NetHack will punish you for killing your pet, but not for abandoning it to starve. This may be somewhat justified by the fact that abandoned pets don't actually starve to death in NetHack — if a pet would've starved while on another level than the player, it just reverts to a non-tame (friendly or hostile, depending on how long it was abandoned for) monster.

One advantage of leaving your initial pet behind instead of killing it is that you can come back later and re-tame it with food if you later decide you want it after all (e.g. for a shoplifting accomplice). Alternatively, once the former pet has reverted to a non-tame state, you can come back and kill it without repercussions (e.g. to loot the saddle off your starting pony). Abandonment on a dead-end level that you don't plan on revisiting (e.g. Mines' End) is also a safe way to dispose of unwanted dangerous pets (like a tame lich) that you might end up with as a result of polytrap encounters.

If you insist on killing your pet while it's still tame, the safest way by far is to let someone else do it. The tricky bit here is that simply leading your pet to a tough monster isn't usually enough, as pets will refuse to fight monsters that are too dangerous for them. However, there are several ways to work around that:

• Let an intelligent monster with equipment do it. Pets don't account for the opponent's equipment in their danger assessment, and many a dog or cat has paid fatally for the failure to tell the difference between "a dwarf" and "a dwarf with a mithril-coat and a +3 mattock".

• Wear a ring of conflict, if you have one. Note that this will also cause your pet and any other nearby creatures to attack you! Still, if used carefully, this can be a very efficient way to dispose of pets.

• Maneuver the pet to be between you and a monster with a ranged attack. Of course, you need to make sure you can handle the attack if it hits you instead of your pet.

• Maneuver the pet into a trap. The big risk here is that killing your pet by displacing it (i.e. by forcing it to move by stepping into its square) into a trap or water (or lava) carries a much worse penalty (angry god vs. just −1 luck) than just plain killing it directly. So, you know, just don't do that, unless you like having your prayers rejected. However, if you can instead, say, lure your pet with tasty meatballs into the path of a rolling boulder, that should be just fine.

If you have the patience, you could also just let your pet starve to death while on the same level as you. However, note that starving pets will become confused from hunger, potentially causing them to attack you, before actually starving to death. Generally, I can't think of any reason to deliberately do this instead of just leaving the pet on another level.

Anyway, before getting rid of your pet, you might want to consider the many advantages they can have:

• Pets can be used to detect cursed items, which makes finding good armor to wear much safer and easier.

• They can be trained to steal items from shops.

• They can kill monsters for you. Especially for "squishier" roles like healer, letting your pet fight for you and level up faster than you can be an excellent early-game strategy (since monster difficulty is adjusted based on your level, but not your pet's).

• They can serve as walking meat-shields. Even if you're not relying on pets to fight for you, displacing a pet between you and a dangerous monster can be a very effective survival tactic.

• They can sweep the dungeon around you for mines and other traps.

• If and when you find a polytrap, you can use it to turn your starting pet into something much more effective. (Wands or potions of polymorph can also be used for this, but they may be better saved for other uses.)

There are several items and tricks that can make pet management much easier:

• Carrying a meatball or a tripe ration in your inventory will make (carnivorous) pets follow you much more closely. (Apples and carrots have the same effect on herbivorous pets like horses.) This makes shuttling pets between levels much easier, and makes it more likely that your pets will be close when you need their help in combat. If you ever want your pets not to stay so close, just hide the meatball / tripe in a bag. In a pinch, you can also use it to tame a hostile dog or cat (although almost any food will work for that).

• Test any whistle you find to see if it's magic. A magic whistle will teleport all your pets (on the same level) next to you, making them much more useful and easier to deal with. Some impatient players may want to abandon their pets on an early level until they find a magic whistle (the Gnomish Mines are a good place to look for them), and then go back to retame them.

• A leash can be used to force a pet to stay next to you, but this can be somewhat inconvenient. Personally, I rarely use them, except sometimes as a poor substitute for a magic whistle when moving between levels.

• Once you've acquire telepathy from eating a floating eye, a blindfold or a towel becomes a useful tool for seeing where your pet went and what it's up to. (Out of the two, the towel is strictly more useful, since it has other uses besides blinding yourself.)

You can just disable starting pet by setting 'pettype' to 'none' in options. See guidebook for more details: http://www.nethack.org/v343/Guidebook.html#_TOCentry_39

Another point that hasn't been mentioned by the other answers yet: maybe you have the option safe_pet set to false. This option prevents you from willingly attack your pet when you move on the same square (unless you are blind, hallucinating or confused). Once you set it to true, pets are much less annoying, and, as others have already mentioned, they become extremely useful.