What trinket should I start with, when should I switch and upgrade?

Solution 1:

TL;DR

Warding Totem

• Safe start, no matter what champion
• Gives you Defensive Map Control (sometimes offensive in Jungle)
• Free ward Jumps
• Buy this item at start 95% of the time

Sweeping Lens

• Safe to buy after you've finished your Sightstone
• Consider buying it against stealth champions (upgrade to Oracle's Lens for stealth detection)
• Sometimes a good Idea on junglers if you want an early dragon
• Can be bought if you are playing a Supporter that relies on Bush-Stealth
• Don't start with the item unless you're planning on camping a lane like crazy

Scrying Orb

• Can be used for scouting Nashor/Dragon
• Could be useful for Non-Skillshot High-Range casts (Nocturne, Caitlyn Ult)
• Generally avoid buying the Item in a serious game

Detail

Usually you want to start with the warding Trinket since it's pretty much the safest and most useful trinket of the three at start. It gives you a free ward for either vision or sometimes even a free flash for champions with Friendly-Dash abilities (Katarina, Jax, Lee-Sin etc). When jungling it can also give you vision control for either more safety or better counterjungling.

The sweeping lens is a really strong Item but most of the time it's only good later in the game. A lot of people buy it as a jungler but usually you should avoid buying it at the beginning because if you buy it early the enemy will still spot you before you can clear the ward. The only time you can actually buy this at the beginning is if you play champions that rely on vision breaks of bushes (Thresh, Blitzcrank, Leona) but even there it's situational and most of the time an extra ward is a more decent start. You want to buy this item if you have a Sightstone as a supporter or for additional objective/jungle control as a jungler but that generally happens in the midgame. Another situation where you could buy it potentially is if you are facing Champions that rely on stealth (Twitch/Akali) but remember that you need the Full upgrade for 475 gold to obtain True Vision.

As you've already mentioned, the orb is a rather useless item. You should avoid buying it on Most champions and the only champions that could actually benefit from it are champions with high-range targeted abilites like Caitlyn or Nocturne. But usually it's still a better choice to take a ward on these champions.