Well i feel the kid is very important in the relationship, so if the kid has a crappy time that is going to effect u, but if kid has good time then looks hell lot good on u, when they are asking next time ur going to take there mom out. So bowling is pretty good seeing all of u can play (besides 8 month year old lol maybe)- best thing for taking out single moms in my opinion.
Other options take her to the park something like picnic maybe, so 5 yr old can go play on swings whatever (u already mentioned parks but its good choice). U can have dinner in place maybe like frankie and bennys or pizza hut, that is also pretty good, they provide crayons and notebook usually. Arcade is usually fun for everyone has well. Once u go out on few dates then maybe u can suggest something specifically for kid like u get this big kind of net thing with loads of balls inside it where u sink in...u know what i am talking about lol.