Can I bring my apps from my iphone to a different smart phone?

I have an iPhone, but I think it is almost time for me to get a different smartphone, as I'm getting really fed up with Apple and iTunes. My only concern is, I've spent a lot of money on apps for my iPhone that I use very, very frequently (metronome, strobe tuner, a few games, a flash card review app, an app that gives daily swim workouts, etc.).

I don't want to lose them when I've put so much money into them already… Would I be able to switch those apps to a different smart phone without having to pay for them again?

You can, if it's an Apple product. If you buy a smartphone which runs on Android, you have to buy all your apps again, since it's not the same appstore.