Pauper is a very cheap way to play Magic, 60 card decks with commons only. Some of the top competitive decks only run about $30, and you can build fun decks up with 10 cent cards from your LGS.

For casual play, my favourite is still Commander though. You can definitely build commander decks on a budget (one of the ones I made run me $26) and the format is a lot of fun!

It's not the absolute cheapest, but I recommend trying out commander if you have not already done so. The value of the preconstructed decks is very high as they come with a bunch of useful rares. Some of these decks are as low as $20 on Amazon. Additionally, almost every card ever made is allowed to be played, and nothing ever rotates out like in standard format. It's a casual format so if you don't have a lot to spend on perfecting a deck, many people have several tiers of decks for more fair matchups. If it's just you and your friends, most of the preconstructed commander decks are well matched. Also there are 100 unique cards in each deck, so there is a ton of variety even in one deck and it will take many games before you even see all the cards you have.

My and my friends always liked brewing our own decks so to encourage that and discourage net decking we've come up with our own casual budget format we call Bargain. The only special rules are your maindeck must be $30 or less on mtg goldfish and your leftover money plus an additional $5 can go toward a sideboard. We just check every couple of months whether a deck has gone way over budget and we haven't had the need to ban or restrict any cards yet.

Pauper is the obvious answer. It's unique in that every card in the format is Common rarity, so that mitigates price already. Check out MTGGoldfish's Metagame page for Pauper, which lists competitively viable decks and their price. Not sure how up to date it is, but it's a good starting point!

What do you expect from those games? Do you want to play magic or are you going for a free for all commander experience?

If you're just playing casually I recommend you get proxies printed. That's the absolute cheapest way to play any format you want.

You can print your own at home, or if you want something similar in quality to actual cards I suggest you check out MPC. There's also a really handy website that makes the whole process of ordering from MPC very easy.