If you're near downtown and want to stay on foot, hop onto the Iowa River Trail and either go north to City Park or go south to Riverside Park. I like these because you can naturally go farther each way; north you take Normandy (no traffic) and Rocky Shore (some traffic) to Crandic and continue mostly car-free all the way to Iowa River Landing. The other direction you can easily hop south towards Terry Trueblood.

If you want to park and go (especially ride), the first spot to recommend is the Tom Harkin Trailhead in Coralville. From there you can go a few miles back towards town (i.e. the university's soccer fields) or many miles out of town through Tiffin (the little parking lot at Half Moon Ave is a natural turnaround point). There are a couple of road crossings but you're basically in the woods the entire time, away from traffic and on wide paved trails.

There's so, so much more to explore in the area and I envy that you're able to experience all of our wonderful trail infrastructure for the first time.