ra = rosin activated
rma = rosin mild active (probably a bit weaker than ra?)
These two depending on the concentration and composition sometimes leave a layer of brownish stuff after you finish soldering (the rosin) which can sometimes be somewhat conductive (or... well, let's say if you work with high voltages this left over flux could decrease the resistance between two wires and cause electric arcs). In general they don't need to be cleaned but they can be easily cleaned if neded with isopropyl alcohol or other electronic circuit safe solvents
no clean = a flux different than ra or rma, that can be left on the board without having to clean it (it's usually not conductive, leaves minimal dust or other crap etc) ... it can be cleaned if you want with isopropyl alcohol or other substances but you don't have to. Don't know if it's stronger or weaker than RA or RMA
Stay away from water soluble fluxes, from what I heard (but I admit I'm not so experienced in this area) they're strong fluxes but HAVE to be cleaned from leads and boards after soldering and it's often hard to completely clean such fluxes if you're amateur and in time the fluxes can do damage.
I use solder with no clean flux, in 2% concentration, bought a 500g spool a couple of years ago), it's Multicore 0.56mm 63/37 Type Crystal 502 , here's the exact link :
http://www.digikey.com/short/344qdcI'm very happy with it. 0.56mm is just right for most soldering, 0.51mm would also work just fine..
If you go with thicker stuff, it's harder to solder surface mount stuff.
If you go lower at 0.39 mm or something like that, it may help you if you plan to do a lot of surface mount soldering, like soldering stuff that's lower than 0805 resistors or capacitors but when it comes to bigger stuff like large through hole capacitors or to-220 parts, you would find yourself having to feed a lot of solder wire into the area you want to solder so you don't get such fine control over the solder quantity and in addition, you would find the flux inside the solder often not being enough when you solder big things.
Sure, a simple solution would be to cut two strands of solder wire from the spool and twist them together to get a thicker strand of solder wire but it's annoying to do this over and over again when you have to.