Is there any other way to heal hearts (damage) besides paying 1 gold in town or by getting the right cards? Can you rest and heal hearts?

no but runebound is a flexible game so you make your own house hold rules. on a website i heard they had a variant where you keep certain movement dice away from the others (similar to healing exhaustion) and roll those first, count how many symbols match your space and thats how many hearts you get, then continue with the rest of the dice that you were supposed to use, but this was initially made for solo runebound, and i personally dont use it but if runebound is too hard for you use it.

As the Shadow stated, there is no way to rest and heal hearts in the base edition of the game so you'll need to have your hero travel to a town, use equipment or ally special ability, or earn healing as a reward for a challenge.

That said, Runebound is a game with many, many fan made variants. Some make the game easier to help out those still learning the rules. Others (such as many designed by myself) intentionally make the game much harder to give experienced players a hearty challenge. The trick is just figuring out what makes the game fun for you and your group. As for healing variants, I often teach the game to new players, and when I do, I use the following.

Healing in town.

Pay 1 gold, heal all fatigue from your hero and party.

Pay 1 gold to heal 1 wound from any hero or ally.

Pay 3 gold to heal all wounds from your hero and allies.

Healing in the Wilderness.

If your hero does not end their turn on an adventure gem or city space, they automatically heal 1 wound.

Judd

In the base game those are the only ways, but if I remember correctly there are extra items or spells in some of the expansions that may allow you to heal in other ways.

The notion of healing in the wilderness is a nice one but I think a bit too easy. I might go for missing a turn (if not gem or wilderness encounter occurs) to heal 1 life for the Hero and Allies. That sort of fits Armstead's original question, keeps the competitive pressure on, and doesn't cheapen the hard won items and options for healing on the go.