As someone who has gone to watch a lot of mecha-anime (surprisingly not because of Armored Core 6. Need to check the franchise and game out soon) lately here's some I've liked. Some I've seen recently. Others many years ago:
1. Universal Century Gundam - Currently going through this, but for example Origin was amazing, and Zeta is extremely promising. Undisputed classic of the genre focusing on the war between space-colonies and their new, independent government and Earth's Federation wanting to keep their foothold in these colonies. Original Mobile Suit Gundam (0079) hasn't aged all that well if you ask me, but it's worth experiencing in one form or another if you want the full UC-story. Going against every Gundam-experts advices I myself recommend three compilations movies at first, and then return to watch the original TV-series when/if you've gotten more into Gundam (and those wondering: Yes. Seen both TV-series and movies).
2. Mobile Suit Gundam: Witch From Mercury - One of the many AU-Gundams, and only one I've seen so far out of the AU-series. Has it share of more serious themes, but wouldn't consider it to be as serious as UC-Gundam is. Lot of nice characters and the animation is good.
3. 86 - Another more serious mecha-anime. Tackles the themes of racism and child-soldiers sent to their certain demise. First season I think was better than the second one, but I do think that this is probably the best mecha-series to come out recently.
4. Gurren Lagann - More over the top-one. Excellent animation and simple message told in a flashy way. Has it's serious moments, but does focus more on being fun and over-the-top in my opinion.
5. SSSS.Gridman/Dynazenon - Another one that's not-so-serious. This time kaiju's are the enemies. It's from Studio TRIGGER so it looks amazing and animation is top-notch.
6. Gunbuster and Diebuster - Honestly I can't remember much about these other than liking them. Hideaki Anno aka. the man behind Evangelion was responsible for the Gunbuster, and remember it being more serious out of the two. Diebuster was alright sequel, but the tone was different from what I can remember. Gunbuster especially is often considered a classic OVA.
I also second Patlabor (though I've only seen OVA at the moment. Need to watch movies and TV-anime) and Code Geass, though I usually recommend it for different reasons than majority of the other fans of the show.