Magic: The Gathering

You can make a cheaper deck if you don't mind not winning consistently. Throwing together a fast burn deck should be a bit cheaper.

But I love surprising my opponents more than I love winning, so maybe don't listen to me.

If you want to be competitive on a budget, standard is definitely not the way to go. You can always play jank but if you play against other people they are very likely to destroy you.

Personally I always feel like Pauper is the answer. Lots of variety and reasonably priced decks. The problem is finding other people to play with you, they don't usually like playing with commons only. I own an out of date pauper battle box and it's a great way to play with friends. 500$ is enough to buy a lot of meta decks and the cost per set is usually very low, it's all commons after all. My average per set is 5-10 bucks.

Don't play constructed

For 2vs2, that's too small for a fun draft IMO. But it's a fine size for a sealed deck. 4 booster packs per player, have someone with a bunch of lands so any deck can be built and have fun.

Draft needs 5 players minimum to ensure each color has a player. Honestly, 6 is the smallest I ever go. 8 is ideal.

Commander as a few have mentioned is a bit on the cheaper side. Modern, especially competitive, is expensive AF. Commader let's you use pretty much your whole collection as options.

Buy singles over packs (buying packs to hope on getting the card you want is an expensive method). WotC has been money grubbing hard lately so buying boxes, lots of packs, or secret lairs usually isn't the most friendly on the wallet.

Check out "Tolarian college" or "Tolarian community college" on youtube for "is it worth it to buy" in terms of value and good cards from sets. He usually does a pretty good job at it.

All in all, still a great and fun game. You ultimately don't have to spend a ton if you just play with friends and everyone has non-meta power level decks.

Edit: Grammar, spelling, clarity.

Depends a lot on the format. Right now Standard and Modern are very expensive. But if you're looking for something more casual, I don't think there's been a better time to play Commander. Either pick up a precon (which will probably be underpowered but a good starting point to build a deck once you've practiced with it) or check out any of the myriad of Commander YouTube videos or articles on sites like EDHREC. Most areas will generally have a pretty sizeable Commander community you can play with

Can always look into pauper.

Play commander, great format, super fun, not expensive lots of preconstructed decks available