Does it have regular flat pedals? Any kind of shoe will do fine. Even high heels!

I don't know really how you get over being scared of riding alone. I guess you just do it. I never really gave it much thought. I just went, sometimes I'd go out 20 or 30 miles by myself. I'd always take my cell phone, and my husband was/is always willing to come fetch me if I get in a jam. (Then a couple years ago I went 744 miles by myself, but that was after I was a pretty experienced cyclist!)

Your daughter will LOVE riding in the trailer. I'd take my niece when she visited and she'd yell "Faster, Aunt Wachel, Faster!"

As far as what you need, for recreational riding on the bike path you don't need a lot:
Helmet
Water (does your bike have a water bottle cage?)
A flat kit and KNOW HOW TO USE IT (YouTube videos, lesson from your bike mechanic, and/or practice at home). I know they make those patch kits but I prefer a spare tube if I need it on a ride. Your flat kit needs a spare tube, a little pump, and tire levers. This probably goes in a little seat pack that attaches to your seat, and your cell phone & car keys can go in there (if you drove to the bike path)

You mentioned in another post that you are thinking about bike commuting too. For that, depending on what you usually take to work with you, you'll probably need more cargo capacity. A basket, rack & basket, rack & panniers, rack & trunk, something like that. A backpack is ok for a while but you'll soon appreciate a rack.

When you bike commute, you might start off thinking you are a daylight-only, fair weather commuter. Get lights anyway. It's easy to get caught out as it's getting dark. You really need those lights so everyone can see you.