What should I get from the Dota 2 Store?

I would do the same thing here that I do with TF2 and just buy keys and wait for a chest to drop. Or go to tf2outpost to trade for a chest.

Get a couple of International Compendiums for a total of $20, sell both of them for 9 Treasure keys on dota2lounge.com and use the keys to buy whatever you like!

Yeah, get a bundle of gifts that is cheapest than keys and gives you more items. Then Go to a private game with a friend and use it, becuse it gives the gifts to the ppl in the server randomly.
If you are "pro" at dota 2 just add me: Tr0wB3d3r
is my alt acc don't think i'm n00b ;D
Gl hf

Get The International 3 compendium, gives like 225% XP bonus till the end of The International 3 tournament which ends ina bout 2 months, also you are supporting Valve which is great and also supporting eSports and the teams participating in the International 3.

i bought treasure keys THATS THE BEST

"What Equipment, Tools, Couriers, or Announcers should I spend on?"

Since no advantages come from anything ingame, you really can go with anything you like, there is no right choice to make (that is until they give in to the mass demands and release a GLaDOS announcer pack, which will automatically become the right choice for these type of questions), so whatever floats your boat :)

Could I just buy keys from the steam market?

just get nothing

Get a bundle for a character you like. May I ask why you're buying something for Dota 2 if you're not all too into it yet?

a pennant, people will be willing to trade that for 3 rares

Scratch trading I would say. Just get the regular and mega-kill bastion announcers.

Logan's voice... Mmm