They might have been cheap but they are worth MUCH more, as I'm sure you know, so take care.
I have been in the hobby for about two years now and when I started out, anything that needed component level repair, I passed on because I didn't know what to do or how to solder. Like anything else, your approach is key.
My general advice would be - don't go too hot. I used to use a huge old iron that could solder the Titanic. Bad idea - I burned components and lifted traces all over the place. I now use a simple little 40 watt iron with a pencil tip and built in light - works great.
I also use a solder sucker, or pump, as personally I didnt get on with braid / wick.
Get a metallic sponge to clean the iron tip - a clean tip is king. Also I find extra Flux added to the work helps solder flow.
When desoldering, I usually work with solder- side upwards. Don't worry about things falling when loose, they are tiny and weigh next to nothing, no damage will occur. Real damage will occur by applying too much heat so practise on some junk as advised above - really do this, we are not trying to add more hurdles for you
If a component won't come off the board with finger pulling once desoldering, don't use pliers. I haVE done that a pulled off traces from the Board. Go back and desolder more.