Broadly speaking, bike choice is based on needs. However, your needs here vary quite a bit.
3 mile trips around town can be done with basically any bike, and almost any bike* (including road bikes) can be equipped with racks and panniers. Whether you want a road or hybrid or mountain or cruise is a balance of speed, efficiency, comfort and whether you plan on taking the bike off-trail
Taking a bike on trails and loops that are 15-50 miles is an entirely different matter and at the opposite of the biking spectrum. The word "trails" suggests non-paved surfaces. If there are a lot of these I'd avoid dedicated road bikes with skinny tires, but if they are packed earth with not a lot of hills you can use a gravel-grinder or cyclocross in addition to a mountain bike. 50 miles is a seriously long ride for anyone NOT on a road bike (and even then it's an accomplishment). Sure some mountain-bikers cover such distances, but that's a feat and not something you'd casually be asking on forum.
So... where and how exactly do you plan on riding? Think about how often you might be on non-paved streets, how far you want to go, how fast & efficient you want to be and whether there are big hills in your area. If you are taking trips with you young daughter you won't be needing to go particularly fast.
*the exceptions are bikes which do not have the mouting points - often cheap-o bikes from big box stores (bicycle-shaped objects or BSOs) and high-end road-racing bikes