I've worked with Red Cross shelters a lot. For anything over a few hundred people sheltered, I wouldn't expect the shelter to provide anything more than a 2' x 6' piece of floor to sleep on. No cots. Nobody keeps that many on hand. Food should be provided, but be warned that it may be slow and not a great selection. It's hard to feed thousands of people on short notice and on a budget. Animals normally aren't allowed inside, but I've seen a lot of families keep the dog out at their car.

I'd bring bedding, including any air mattress, snacks, possibly whole meals, drinks, games, books, medications, toilet paper, paper towels, etc. It's essentially indoor camping. I wouldn't bring anything too valuable if possible. Security can be spotty at shelters and in the parking lots, especially if people are freaking out.

Boredom is probably the biggest problem I've seen in shelters. Most shelters don't fill up or explode into anarchy, but instead linger on for a week or three with a dozen families or less in them. The kids especially get bored quickly.

If the shelter is crowded, offer to volunteer. That should fight boredom, might get you a better spot to sleep with more space and less noise, and would help others.

I hate to say it, but volunteering might get you better food or at least some variety. Not because anyone buys better food for volunteers, but because people spontaneously donate meals to shelters and the policy normally is that they can't serve it to people (for safety reasons). Volunteers shouldn't eat it either, but it happens.