Of course contact lenses are essentially safe, they wouldn't be legal if they weren't! (that's not to say people never have problems with them, but it's not that common and problems aren't usually that serious). In terms of losing them, no they won't fall out of your eye.
For occasional use, daily contacts would probably work out cheapest, but be warned -- once you've tried life with contacts you really won't want to go back to glasses (or at least that's what I found). For everyday use, go for monthlies as they're cheaper and just as good. And buying them on the internet is cheaper than buying them from a High Street optician, although you should probably get your first lot from your optician to be absolutely sure that you're getting the type suitable for you.
With regards the different types of more expensive lenses you mention, your optician should tell you what type you need. If you don't have astigmatism and are just normally short-sighted the cheapest should be fine.