A little late to the thread, but I'll second MTGO. It's a little cumbersome at the start, but is a great way to really understand the phases, priority, how/when abilities work etc. You can build a deck, then
launch a solitaire game and practice the fundamentals and see if you like the deck and learn how to pilot it before jumping into the queues. Virtual cards are generally cheaper than their paper counterparts (some exceptions since a lot of sets predated MTGO or had limited releases online, so they are scarce.)
I like Modern as there are unlimited types of decks and there are lots of sources out there on how to do it on a budget. Legacy can get really expensive in a hurry for competitive decks. (I guess that's true for any format, but there are lots of really strong budget Modern decks.) Check out youtube or sites like MTGgoldfish for budget ideas.
Pauper is a great way to get started and you can try them on MTGO for a few bucks for a decent deck and 50-100ish for a really strong deck that can win you some tournaments. (lots of expensive cards are in the sideboard, so you can build the main deck for a few bucks and have fun, you just might not have the right sideboard answers for some big decks in the format. (ie, izzet delver is one of the main T1 decks in the pauper meta right now. A tier 1 version is around 100 online or 150 on paper. Of the 100 online, about 80 bucks of that is optimized land and sideboard cards! You could totally have a lot of fun w/o those for $20!) I have a ton of pauper decks online, and I don't see any of mine that are more than $20 and I have a ton of fun with them. I would need the "expensive cards" to compete, but I play for fun. The downside of pauper is that it's hard to find an LGS that has regular tournaments in it, so you may be stuck to online or with friends.
My hope with MTGO was to try a few (budget) versions of competitive decks (in pauper mostly) to see what mechanics I liked, then once I nailed it down and knew how to play it well, "invest" in competitive version of that archetype on paper (and probably Modern) to take to the LGS (or work... lots of players at my office.) Lol, it didn't quite work out that way as I keep trying new ones and have seldom used the paper ones I've built, but I'm having fun!
One thing you can do if budget is super tight is play it a deck online to practice, buy the cheap paper cards for it to start your paper copy, then create an amazon wishlist for the rest and send that to friends/family for gift ideas on holidays. (I actually got a
Karn for Christmas that way... and I'm an adult with teenage kids... I do well, but have a hard time spending money on the expensive cards for decks I'm not competing with) hahaha
People might not like this suggestion, but "net decks" are a pretty good way to learn and play competitively right off the bat too. They are powerful and can compete at your LGS, are pretty well sorted an can be pretty budget. 8-whack goblins, bogles, blistering rage (aka glass cannon), elves, red deck wins, burn, infect, etc. can all be really strong, fun, and built on a budget. Knowing how to pilot your deck is probably as important as your deck itself, and since you can't get out much to practive, def. try MTGO. For the price, I think this deck might be hard to beat: (for paper or online... $1 online, $20 IRL! I've played it online and it beats a lot of decks it shouldn't!)
Black Rogues: Unblockable, UntouchableHere's a good primer on how to get going on MTGO. Add that rogue deck and go kick some butt!
https://www.youtube.com/watch?v=PQRu9Cts2r4 Have fun!