For those of you unfamiliar with it, Super Robot Wars is a turn-based strategy game that, with a few exceptions, take a whole bunch of mecha anime, mixes their plots, characters and machines together into one world. So you'll have, for example, Gurren Lagann from Tengen Toppa Gurren Lagann fighting alongside Wing Gundam Zero Custom from Gundam Wing Endless Waltz and Evangelion Unit-01 from Rebuild of Evangelion.
Most of the series is only in Japanese due to licensing issues, but the most recent batch of games have had official English translations for the PS4 are available in the Korean version - and since the PS4 isn't region-locked, you can either import the games directly, or purchase an Asian Playstation Store card and purchase the downloadable version with an Asian Playstation account. Many earlier games also have English fan translations as well, though I won't say how you get those, lest I break the forums' Terms of Service.
One of my favorite elements of this series is that having the presence of other characters can change how the events of a given series plays out. For example, Shinji Ikari tends to end up being less of a whiny kid due to being surrounded by people who serve as his friends, and ends up doing some really manly stuff.
If you're looking for some games to try, here are my suggestions (all have Super Robot Wars/Taisen in them, and all have at least partial translations, either official or fan):
-V (PS4) - the first Super Robot Wars game with licensed characters that has an official English translation.
-X (PS4) - the second licensed SRW game to get an official translation into English.
-Alpha Gaiden (PS1) - This one is notorious for being really, REALLY hard. Full fan translation.
-J (GBA) - This one's pretty easy, making it a good introduction to the series. Full fan translation.
-Original Generation (GBA) - Only one of two turn-based strategy SRW games to be released in America. It's a good game, but had zero marketing in the West, so it didn't sell well, although it did get a sequel on the same system.
-W (DS) - This one is definitely one of my top favorites, but this one also has VOLTRON.