I saw Terminator Genisys last night. Fortunately, Karen prepped me for it ahead of time, so I thought I'd pass the favor along by telling y'all my recommendations for how to approach it. No spoilers here.
First, if you haven't seen the original Terminator in the past year or so, go rewatch it before seeing Genisys. About 3/4 of the best things about the new movie are the bits that revisit/rework/revise/call back to the first movie.
Second, if you liked T2, you might as well rewatch that too. The new movie doesn't do as many specific callbacks to it, but there are a few, and there are definitely general things about T2 that are important to remember during the new movie. Also, if you loved Linda Hamilton's shoulders in T2 (nearly everyone who I've ever asked what they liked about T2 has said “Linda Hamilton's shoulders”), you might as well get that out of your system, because her shoulders do not, alas, make an appearance in the new movie. Nor does the rest of her.
In order to enjoy the new movie, there is no need to rewatch anything else. For example, there's no need to rewatch T3 (although I—unlike most people—rather liked it, and am looking forward to seeing it again at some point), and sadly no need to rewatch Sarah Connor Chronicles (although it couldn't hurt to rewatch the first episode, to which I think there's a momentary callback in the new movie) (and for that matter, it couldn't hurt to rewatch the whole series and to sigh about how sad it was that it got canceled). There's definitely no need to rewatch Terminator Salvation, which honestly I remember almost nothing about.
Okay, I got a little carried away there. My point was meant to be that having T1 fresh in your mind is a good idea, but you don't really need to rewatch anything else, and I think the new movie takes only the first two movies as canon.
So much for what you need to rewatch. In terms of mental-state preparation, I recommend being prepared for some miscasting. Two of the leads of the new movie aren't really right for their roles; they don't look or behave like my idea of the characters. (A third lead took a while to grow on me, but eventually won me over.) However, Arnold is pretty much perfectly cast, and the majority of the best moments in the new movie revolve around him.
I guess the last thing I would say is: Don't have high expectations. This new movie has half a dozen fun bits, some nice (though plot-holey) time-travel stuff, lots of nostalgic references, and some pretty funny lines; but imo it's not that great a movie overall. Despite a few really good lines, most of the dialogue is a little weak. It has some structural issues, it has some major plot elements that left me thinking “But why didn't they do this other thing?”, and it's certainly nowhere near as good as the first movie. But if you go into it in the right frame of mind, it's sufficiently entertaining.