You can see a Discovery shuttle in this trailer.
I've seen a few people saying this but not sure I understand why it's a problem. In both Children of Mars and this trailer the shuttles are being used for civilian transport. That...kind of makes sense? Assuming that the Federation builds ships to last, shuttles that were fit for service in high-end Federation ships in the Discovery era should, with a bit of retrofitting, still be okay for civilian transport in the Picard era. It's a long time (~140 years) but not a totally unbelievable amount of time. There's still amphibious vehicles that the US built in WWII in use for tours today.
Why are we seeing a Galaxy class at all? Is it just meant to be a memory of the Enterprise?
The Galaxy class is less than 40 years old. There's no reason for it not to be around. There's active ships older than that in the current US navy. It'd presumably be considered outdated, but for a lot of what the Federation does it'd be perfectly fine.
To truly understand Picard's psyche going into the new show, one needs to see at least a few Lwaxana Troi episodes.
This is a better reason than I expected for posting that list of episodes.
But still no.
Do we know if we'll be able to go into this blind? I really like Star Trek Discovery, so I'm interested in this. I've saw enough back in the day to know the basics about Next Generation... Data is a robot, Jeri Ryan was a Borg, Riker was Picard's #2, but that's about it.
I think it'll be possible to go in blind and pick up most of the key points along the way, but probably better to have some background. There's viewing lists online, though there's quite a few episodes and movies on them and it might be tricky to get through those lists before the first episode of Picard. If you want the quickest catch up it might be best to go for the TNG Borg episodes - "Q Who", "The Best of Both Worlds" which is a two-parter and "I, Borg". Those four episodes give a really good look at Picard as a character, the Federation as a polity, the Borg as an enemy and the crew of the Enterprise. First Contact is an action-focused addendum to that but I'm not sure it'll be a required watch.
Some guides suggest watching Romulan episodes of TNG as well, and although there's some very good ones, you will probably be able to get by with just a basic summary of the Romulans (proud people, Star Empire, capital planet recently destroyed by a special effect). Previous to that, the entire Romulan leadership was wiped out (by an entirely different special effect) so in terms of knowledge of how the Romulans are now, we're all pretty much blind.
I also see that Nemesis is appearing on some suggested catch-up lists...don't watch Nemesis. Read a synopsis on Wikipedia or something. It's not worth two hours of your life.