Pretty much will echo what was already stated.
The Era Digests/Reports should get you moving along with some rules tweaked for the respective eras.
3052 covers from the 4th Succession Wars to the Clan Invasion era character building.
Tech Readouts are useful and if you're looking for specific cultures to pick up on, the old Housebooks can work (though they are now pretty much obsolete given how we have Handbooks for the Houses).
Though the listed TROs in other posts are useful, there's also the TRO: Succession Wars which covers a lot of the Mechs as well, but I think it misses a few chassis here and there. Plus for vehicles, it's probably better to get PDFs of the ones where the years are listed out.
Chaos Campaign Succession Wars is a free download which is a nice abstract way to get the shooting going.
Record sheets, I really recommend looking at the Forcepack pdfs in the download section of the site:
https://bg.battletech.com/downloads/Yes some of them are Clan Invasion era, but it's fairly easy to see what is where. That whole Kickstarter was a blessing for the Succession War and Clan era when you consider how many sheets are just there for free.
From a non 'Mech centric angle, most stories or scenarios from anything can be tweaked for a Battletech flavor, but I do know there's some Free RPG Day scenarios for AToW online.
Shameless plug here: Shrapnel, the magazine that gets released every quarter, sometimes has a few RPG scenarios you can adapt for your tables. A few of the authors enjoy trying to put more out there for the RPG when possible. Just read up on the reviews from the others in the forum when they release to see if a specific issue is worth a purchase for your needs.
From there its just up to you and the players on where you go.