I'll respond more directly, because I don't want you ending up like me before I learned to build, and buying one weapon, one armor, one pair of boots, and then potions for the rest of my inventory because I thought I was done =P
Boots and 3 potions is a good start, and one used commonly at high levels of play, because it provides both mobility and sustain, meaning you can dodge abilities, position yourself for last hitting and harassing, and survive longer before going back.
From there, you want to start building something that will define your role as a champion, and modify it as needed for the current game.
I'll give you six very basic, general build strategies, and leave the specifics up to you as you play more and become more experienced.
A support build will want gold generation items, wards, and items that can benefit your entire team, like auras and actives. You'll generally start with a faerie charm, several wards, and a potion or two. The charm gives you a start on the gold generation item Philosopher's Stone, and provides sustain for your mana based support abilities(which most support champions uses). From there I usually build Heart of Gold followed by boots of some kind(Ionian for cooldown reduction or Tabi/Treads for defence are generally good options). After that I'll build an Aegis of the Legion, and maybe a Soul Shroud or a Zeke's Herald, depending on which I think will benefit my teammates more. Finally I'll build my Philo stone to a Shurelya's Reverie(Move speed activate) and a Randuin's Omen or Locket of the Iron Solari. Some supports will rush Shurelya's earlier in the build. You probably won't use this one much at your level, as it's rather useless against bots.
The next build, the one that will probably work best for you as Teemo or Caitlyn, is a full AD build. I start this with boots and potions, although some people will take a Doran's Blade. Next, you have an option of scaling with Doran's Blades(usually no more then two), or start building your core items directly, which is probably more effective for AI or low level PvP games. There are lots of ways to do this, so I'll just go over one build for now - grab a Vampiric Sceptre for extra sustain from your autoattacks, or a B.F. Sword if you can afford it in a single back. You should also upgrade your boots to Berserker Greaves as soon as possible. Then go Pickaxe or Cloak of Agility, and eventually build an Infinity Edge from those items. Next I go with Zeal(or its smaller components, Brawler's Gloves and Dagger) and build it into a Phantom Dancer. Bloodthirster and Last Whisper are also good items for this build, one providing large amounts of AD and Lifesteal, and the other granting Armor Penetration. I like to finish with Madred's Bloodrazor to help deal with high HP tanks, but you can also add a Guardian Angel for Armor, MR, and the resurrection effect, or a Black Cleaver or Ghostblade if your opponents are building lots of armor against you.
This is getting longer then I expected, so I'll be briefer with the rest - there's full AP, where you'll want Magic Penetration, Ability Power, and Cooldown Reduction; full tank where you want armor, MR, and health and health regen; AD bruiser where you want AD but also durability, going for items like Frozen Mallet and Wit's End; and AP bruiser where you want AP and durability.
There are many, many ways to combine items to achieve the results you want, and it's also worth considering certain specialized items, like the Quicksilver Sash, that have activations that can be very useful in some situations.
Hopefully you have an idea of build order from my more detailed paragraphs - if not, you basically want to prioritize the items you pick up to be most effective and help you scale through the game, which you wouldn't do well if you just went for an item like Warmog's armor right away.
As you play and become more experienced, you'll get a better idea of what items you want to build for what uses, and when to build for a specific stat.
As a final tip, stacking any item other then Doran's items is usually not going to be worth it.
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