Hi Abefroman,
Actually, I'm also not crazy about too much bitterness (I don;t like chicory) but i do love rapini.
I think the rapini I've tried in the states are stronger than those you can buy here, and probably partly due to the soil, maybe the particular type, but probably mostly because they're harvested too late. The stems should be the thickness of pencils, not thicker, and the tops should be mainly like dense buds, not open and loose - they're about 1/2 inch across - maybe bigger, but still tight. The leaves shouldn't have really vibrous parts that don't cook easily. So if you do decide to give them another try, wait till you find them that look tender and less mature. (Here they are a winter vegetable, can;t find them in the summer).
The fact that you ask if you eat the leaves tells me they must be too mature, because rapini are almost all leaves, like spinach, but with little buds in the middle - if they've grown further, the buds become predominant, and then they would, i think, become more bitter.
Then, they take some preparation - you have to peel the stems (a real pain in the neck - here you can buy them by the kilo already peeled and picked through, sometimes already having been washed once to get rid of most of the sand).
People do buy them by the kilo, and a serving is usually half a plate full.
Boil them in plenty of salted water till just tender, (did you undercook them?) drain, then heat olive oil and some sliced garlic and if you like, hot pepper flakes. Let the garlic begin to soften very slowly, then add the rapini and sautee till all of them are well seasoned.
Of course, you still might not like them.
I came from not liking any bitterness, to liking a little - stewed belgian endive, for example. Rapini, i would say, should be about that level of bitterness. If they're stronger, they're probably left too long before harvesting. But you still might not like them, we won't hate you for it. Not everyone likes bitter. (Some people drink coffee black without sugar - now that's bitter, and I'd go without coffee if i had to drink it like that. And lots of americans drink espresso without sugar! eek. - you'd be surprised how much sugar italians usually put in their tiny espresso - 2 heaping teaspoons is not unusual. So, taste is personal, and unless you have an italian mother in law who is cooking rapini almost every day, you can live perfectly well without ever eating them.)
Oh, and also, you don;t eat italian food without bread. Rapini are eaten, like practically everything here except pasta, with bread - pretty much bread in one hand, the fork in the other. That kind of tones down the bitterness.
And one of my favorite suppers is a nice big dish of rapini and some cheese and bread. They go really nicely together, the cheese balancing the rapini and the bread smoothing everything down together.