I've just started looking at the Big Island, and I think we would definitely want to spend some time in Hilo - it seems like our kind of town. Is it possible to stay somewhere that we wouldn't need a car to get around, at least for a few days? Are there busses at all?

I will be with my husband and 5 year old son. We've never been to Hawaii before.

I would definitely recommend getting a car. It's called the "Big Island" for a good reason. As for Hilo, it is the 3rd largest city in Hawaii and is really spread out.
It all depends on what you want to do. Banyan drive is the area where the majority of the hotels are clustered. There are a few restaurants and parks in that area. It is a few miles to the historic part of Old Hilo. And a few miles to the shopping malls and movie theaters.
There is limited bus service in town. I am not sure of the routes. I think the bus is still called Hele On.

You need a car. The town is fairly small and walkable but after a few hours you might have seen it all, so you'll want a car to get out of town and see the waterfalls, the botanical garden, etc.

Hilo reportedly has several good B&B's, if you're interested.

Hilo is a great place. The Hilo Hawaiian Hotel right on Hilo Bay is a nice place to stay. Get a room with a lanai (balcony) facing the bay. There's also a swimming pool.

I agree with those who say rent a car. You are actually closer to Hawaii Volcanoes National Park in Hilo than on the Kona coast, but you still need a car to get there. There are also waterfalls to visit on the n Hilo side of the island.