I play mostly standard, but like to draft whenever I have enough gold or gems to do so. Be aware that the Historic format in Arena will include Alchemy which has some digital only cards and mechanics that don't exist in paper. Some people like it, I don't care for it.

Be sure to redeem all the codes for free packs before spending wild cards.

I play historic brawl. It can be slow to get into for someone new to Arena, but here are some ideas for budget decks: https://mtgazone.com/budget-historic-brawl/. Once you've been playing for a while, historic brawl is the cheapest, least wildcard intensive format. That's because it's singleton and non-rotating. I also like the randomness that 100-card singleton provides.

I should mention, however, that it's not as similar to edh as you might expect. For starters, it has a smaller card pool, 25 life instead of 40, and no commander damage rule. What makes a bigger difference, though, is that it's 1v1. Effective historic brawl decks usually pack more interaction, because you can't rely on the table to keep an opponent in check. Some people start historic brawl expecting to be able to use edh-style decks that build their position slowly or focus heavily on the commander without much protection for it. They are quickly disappointed, because those decks don't work nearly as well in historic brawl. Control is prevalent, aggro is fairly viable, and if you want your commander on the battlefield, be sure to run things like Slip Out the Back, Tamiyo's Safekeeping, etc.

I think brawl is the best if you're looking to be effective with wild cards. It's a less competitive format that you'll be familiar with if you have played edh, and you will spend less wildcards solely because you can only have 1 of each.

As for what I play, when I started I crafted the best mono red standard deck I could and farmed my 15 wins of the day until I could afford to build any brawl/standard deck i wanted with my wildcards. But that path requires you to really like playing red.

Historic is expensive and the only format I do not recommend.

Depends on what you're looking for. Do you want to play competitively and gain ranks for the season rewards, or just have fun with it?

For just having fun with it I'll echo the recommendations of Brawl (probably Historic Brawl). For competitive, Explorer is probably the safest bet. It's a bit more competitive than Standard, but Standard is pretty high power nowadays anyway.

Standard and limited for me.

I usually play Historic Brawl, because with the commander and the 100 card deck limit it's the closest thing I have to paper commander while I don't have any room to play paper commander!

I do sometimes play other formats like limited quick draft or other constructed formats in order to finish off my daily quests

Mostly standard, since I don't like drafting to much. And due to the not rotation I finally have a solid land base to try a lot of different stuff.

Historic Brawl is my go-to. It's closest to EDH. Being a crazy Johnny player, I'll throw together a standard deck every once in a while if I see something with interest interactions on a CGB or Legend video.

90% of the time I'm playing Explorer since I like my decks always being available, and I'm not a big Alchemy fan, but I do dabble in Historic/Standard/Historic Brawl on occasion.

Red aggro decks would be a good starting place, they're good enough to keep you playing and cheap enough that you arent burning tons of wildcards on most decks. Then as you build up your wildcards and test out other modes you'll have a better starting place.

I enjoy Explorer lately although the meta is very stale some days.

Otherwise Historic Brawl is fun. When I feel particularly casual I just skip opponents on there if I stare down the 6th Atraxa pile in a row.

Also, for me Explorer feels nicer than Historic lately due to the digital only mechanics taking hold in the meta. Just not my favourite style of play.