Dies to Removal

Do you have Cockatrice? If not, download it, then load up a few decks and test with them. I used to do this all of the time, just spend some time reading a primer online, then throw together a list on Cockatrice and mess around with it for a few hours. For starters, here's my infect list:
http://tappedout.net/mtg-decks/dirty-pump-infect/
I think there's also a primer for RDW on this site, as well as on MTGS.

You also have many other options for getting into Modern. I was able to start out very cheaply with a mono green infect deck I had from Standard. During Standard season, I think I only put $20 into the deck. When my LGS announced that it would be running Modern FNMs, I spent another $20 on upgrades, and it's done fairly well so far. Sure, it could be much better if I put a few hundred dollars into fetches, shocks, and Hierarchs to make it BUG, but I'm not really interested in that. Instead, I'm accumulating store credit from FNM so I can build a deck that I would really want to play, like burn or RDW. You could build infect for cheap and use it to win cards for your "dream deck", then trade the cards away for r/w cards when you're close, or you could start with a simple version of what you want to build, since then you're already part of the way to completion when you start, and you'll have more practice
with the deck after you've built an optimized version. For you, I'd recommend building mono-red with only mountains, as many of the cards that it plays are either very cheap, so you won't care be losing much once you cut them for white cards; format staples, meaning that it'll be easy to trade them for upgrades; or cards that you'll also want in your Boros deck.

I'm not sure Boros is your best option on a $75 budget. The successful Modern Boros lists tend to be Landfall decks, which rely heavily on fetches. However, without fetches, which would cost at least twice your budget, building this is not possible. The deck tends to be a red deck at heart, with white splashed mostly for sideboard options and Path/Helix. On a $75 budget, I don't think you'd be able to sure up the mana base enough to make the benefits of adding white outweigh the complications. That being said, while the deck might not be the most powerful when you've only put $75 into it, if you continuously invest in it for a few months, it'll get there, and you'll still be playing something that you like until you get there.