I can answer some of that for you:
Amazon - I'm in the US, but I believe you would sign up through the "Associates Programme" link on Amazon.co.uk.
Ebay - They've gotten more selective about their program than they used to be. You do need a site to get into the program, and you need to have more than just a shop full of their links. They'll want to see related content. (Search engines will want to see that too.)
The type of advice I'm looking for is practical advice on how to build an affiliate site and what domains are worth doing so.
I highly recommend spending some time on the ABW affiliate forum I mentioned - read, ask questions ...
As for what to start with - pick something you know about or are interested in learning about. Done right, affiliate marketing isn't about just putting links and banners on a page and driving traffic - that may work for Adsense, but it isn't an efficient way to get sales. Think more along lines of guiding your visitors into making a purchase decision. You'll be more effective at this if you know your subject.
Scope your topic out for the amount of competition, the quality of the competition, and the availability of affiliate programs with a good reputation in the aff. community (i.e. the links track, they don't have a high reversal rate and they pay on time). Look at the prospective merchant's site - would you buy something there? If not, your visitors won't either.
Amazon and eBay are reputable and familiar to people - sometimes that works in your favor, sometimes not. Neither pay all that much - eBay is PPC with the amount you're paid based upon previous auction wins resulting from clicks from your site. This can be as little as 2-3 cents/click (and they won't pay anything if there's a long streak of no wins or BIN's). They have been known to drop affiliates from the program for poor performance.
Amazon has a fairly low commission rate and a short cookie duration - if the person doesn't buy immediately after they click through from your site but returns later and buys, you may not get credit for the sale. They aren't likely to drop you unless you violate their program terms.
Build a site with some content first, then apply to the program(s).
Basically I want to create a lot of affiliate sites that would earn 10-50 USD a month each.
You're better off building one site to start - get that going then build another one. It's easier to grow one site to several hundred or more a month than manage a bunch of "maybe it will earn $20" sites.