Dont know if anybody already recommended it: Patlabor, both the series and movies. Police mech for the win!
Seconded. Franchise needs some clarification however. Anime has two different story lines. First was OVA series of 7 episodes, which are followed by 3 movies. Then is TV series, 47 episodes long if I recall correctly, which is followed by OVA of ... 16 episodes. TV series and OVA following it (Patlabor: The New Files) have different story from OVA & movies. All of them should be available for sale either on DVDs or BDs (Blu-ray Disc). I recommend you to start with the 7 episode OVA and buy more if you like it. As noted in a post before, Patlabor is about mecha police doing police stuff, NOT MechWarriors fighting wars.
Other mecha shows of interest for BattleTech players:Armored Trooper VOTOMS
Main character is renegade mecha pilot fighting for his survival against conspiracies, gangs, corrupt police, civil war... Smaller than BattleMechs, mecha in this anime are quite believable, something that could be almost made right now if governments would put serious effort to it. One of the top choices for BattleTech fans.Mobile Suit Gundam
Many series and OVAs on this one. Look up site Crunchyroll, type gundam to search field, and you'll get number of series to watch (for free). Over there I recommend Mobile Suit GUNDAM: Iron Blooded Orphans. It is about child soldiers working as mercenaries. Number of people on this forum swear this anime has got inspirations from BattleTech.Konosuba
Fantasy comedy about Japanese shut-in otaku, who gets called by goddess to save a fantasy world from demon king. Japanese boy is looking forward to grand adventure, the like in (Japanese) video RPGames, only to get his rear kicked by harsh reality (repeatedly) to comical effect. His all-female adventuring party have crippling specializations (crusader is tough as a wall, but can't hit a wall) which makes the guy say: "My party can't be this dysfunctional"
(in translated light novel). Anime is available on Crunchyroll. Don't eat nor drink anything when you watch it
This should get you started. For more anime, check out the anime topic.