I think the city in the game looks amazing, they've done a great job on the graphics there. My character though, well she doesn't look as good as Saints Row 4 and I think that's partly due to the weird lighting effects and the skin textures. Sometimes my character has a wet skin look when she doesn't have that option selected at all from the character creation/editing and it's strange. (The time of day affects this too and what colours are coming from sunlight, so at times it's not so bad.)
Another thing that's really bugging me at the moment is that I played for a while to unlock Kinzie as a homie but I can't call her, she's always busy. Now all of the others seem busy too and on reddit someone said this is an old bug carried over from the original. It's insane if they don't patch that because we're losing out on being able to play with them and I think that's a major bug really.
This is my first time playing SR3 too, I got into this series from Saints Row 4 where I have the game on Playstation and PC, at the moment I only have this on base PS4 so maybe that partly explains character graphics issues.
A couple of other good things so this isn't almost all negative : I thought I'd really miss the freedom of movement from SR4 (a big reason why I got SR4) but I haven't missed it that much at all. You get stuff to fly around in really early on and I'm enjoying driving around, which obviously isn't done as much when you have modded superpowers for SR4 and can jump several blocks away at a time and also manipulate the gliding effect so you can fly across the entire city. So that's also great (and not missing Zin propaganda all over where ads should be too!)
It looks like there's loads to do as well and like I'll really enjoy playing this for a good while to come, but I'd really like to do that with homies that I can call up too, so hopefully this issue either passes or it's fixed in a patch (I know you have to unlock this option too and have had them at times, have been stuck for a while though now with none. I'll post back to quickly mention it if it passes.)
edit : I could recently call Oleg, Zimos and CheapyD, but still no luck with Kinzie, Shaundi, Pierce or Voila. Maybe that'll pass at some stage or if it's a big it'll be fixed. My next mission is Stop All The Downloading for 3rd stronghold so maybe it's something to do with the stage of the game too, but I've not once been able to call Kinzie yet so it seems off.