Against a Melee...
Early you will want to take your E and try to poke as much as you can but still getting your CS and also still getting level 2 faster then your opponent. Once you get level 2 go hard using your auto Q auto combo. Your opponent may be low enough to go back and if so push out the lane and go back using teleport to get back fast and get that CS and exp advantage. Your early game is all about farming for that mid to late game where you can one shot most of the enemy team. You will want to keep up wards to keep you safe until you hit level 6 at least.

Against a Ranged...
Similar to dealing with a melee you will want to take E first as it helps with CS and also some poke when trading. You HAVE to get level 2 first here and trade with them effectively whilst keeping up your farm. Take Q at level 2 and use your auto Q auto combo to take a chunk of their health. If you get out traded be careful and keep up your wards. Simply let don't let the wave push to your tower and back when you can to teleport back in and keep at it. If your opponent has not backed yet use this to your advantage with the item gain and bully them out so they miss CS and exp.

Mid game for a Teemo is all about farm farm farm. You have to farm up to get big. Use your teleport to go bot when they are pushed in and with your Q u should easily be able to very quickly kill the ADC with an auto Q then (sheen proc) auto again. Once you have sheen and you see a chance to teleport then do it. Your damage output now is massive.

Roaming like this will also stack your Mejai's and you will be a late game beast with 800+AP and those Lich Bane Q auto attack proc's will hit for 2k-3k health and tear through basically anyone even their tanks.

Late game you want to be solo pushing and let your team know this as well because you can easily teleport in and turn the tides of a team fight.

If you are pushing hard you should be able to pull the attention of 2-3 of their team members and get your team to take objectives when this happens. Not many, if anyone, on their team will be able to 1v1 you at this point. If you push top or bot hard get your team to go mid or the opposite side and just keep pulling the attention of the enemy team to you.

In team fights you will want to be focusing their squishy champs such as ADC and mid etc. This may sound obvious to most but the ADC has to go first because your blinding Q to him will mitigate half of the enemy teams damage in a team fight.

If you cannot get to the ADC just poke around. Remember that you can proc your Lich Bane by putting down a shroom or using your W speed boost to quickly get in an auto attack and run back. You will be autoing for over 700 damage a hit with your Lich Bane now...