Bulbs are designed to last out over long, hot, dry seasons in a state of dormancy. Usually with a 'coat' of some kind. And with the next season's flowers/leaves already formed within. Think onions as well as garlic.

Some plants that we call 'bulbs' aren't. Like Dahlias. They grow from a tuber, like a potato. And they are damaged by hard frosts - so they need to be dug up and protected over the winter. Or tuberous Begonias. Of the two - the begonias would be more likely to fit in with your gardening preferences.

If you live with a climate that gives deep and hard freezing over winter, or very high temperatures over summer, then some bulbs or bulb types will have to be lifted and stored for added protection.

Other times, you just want the spring-flowering bulbs' space for growing something else over the summer. So you lift the dormant bulbs and store them where they'll be out of the sun, in a cool and shady place, safe from rodents and insects. With good air circulation so they don't go mouldy.

About those tulips you got. They don't do 'indoors' very well. Not for two season in a row. House temperatures can be a bit too warm for them to do their full 'grow cycle' in. Use the Search service to check. There's something about chilling requirements which are needed for continental growers. (Mine stay inground all year, so this step remains mysterious for me:-)) )

They can definitely regrow. However, they might not flower, or you might have to wait two years or so for it to happen. The tulip often produces 'daughter bulbs' (offsets/kiekie) and these have to grow to size - just as orchids do from back bulbs (Cymbidium, for example).

If you are wanting to explore 'inside' or 'tropical' bulbs or bulb-types a bit more - you could look at Haemanthus albi-flos, or Clivia, or even Scadoxus (though it's a bit biggish). And those scoundrels of the bulb world - Hippeastrum, Vallota, Ismene.

There are plenty of bulbs that do not require being dug up every year in your climate.

Crocus, Daffodils, Hyacinths, Iris

Tulips are a bit trickier. Many do not reliably return.

To the above list of bulbs that don't have to be dug up, add Chionodoxa, Pushkinia, Iris reticulata, Hyacinths, Colchicums (fall bloomers). There are others as well.