You are never going to work on a multilayer board as they just don't exist in the areas that you are interested in working in.
Later boards are often quite simple as the components were all inserted from above - usually manually but (much) more recently by machine - and then flow soldered by passing them across a jet of molten solder with all joints being made in one pass.
This can make component removal quite simple because no wires are bent over on the print side of the board. Apply iron and pull component from above or, even simpler, apply iron and solder sucker - if you are lucky, you will end up with a wire in a hole completely devoid of solder.
On manually assembled/soldered boards, the component leads are bent over on the print side of the board. These can be much harder to deal with. However, if you make sure that your iron is fitted with a nice sharp chisel bit - clean and freshly tinned, of course - try this: apply to the joint to melt the solder - it may be advantageous to add a little fresh solder to help it all to flow nicely - then force the flat side of the bit under the wire, between it and the track, so that you can lever the wire end vertical while also apply downward pressure to the track to discourage it from lifting. Once the wire is reasonably vertical, use the solder sucker to remove the surplus solder.
Afterthought ...
For a cheap, non-destructive way of practicing, find or buy a piece of veroboard and make artificial 'components' from tinned copper wire - simple 'U' links - and solder them into random places on the board. There is no need to make the usual track cuts, even if one of your 'components' has both ends connected to the same track!
Try both straight-through (being careful not to burn yourself while soldering them and holding them in place!) and captive, where you bend the wires over after inserting them.
Now practice removing your 'components' ...
Later boards are often quite simple as the components were all inserted from above - usually manually but (much) more recently by machine - and then flow soldered by passing them across a jet of molten solder with all joints being made in one pass.
This can make component removal quite simple because no wires are bent over on the print side of the board. Apply iron and pull component from above or, even simpler, apply iron and solder sucker - if you are lucky, you will end up with a wire in a hole completely devoid of solder.
On manually assembled/soldered boards, the component leads are bent over on the print side of the board. These can be much harder to deal with. However, if you make sure that your iron is fitted with a nice sharp chisel bit - clean and freshly tinned, of course - try this: apply to the joint to melt the solder - it may be advantageous to add a little fresh solder to help it all to flow nicely - then force the flat side of the bit under the wire, between it and the track, so that you can lever the wire end vertical while also apply downward pressure to the track to discourage it from lifting. Once the wire is reasonably vertical, use the solder sucker to remove the surplus solder.
Afterthought ...
For a cheap, non-destructive way of practicing, find or buy a piece of veroboard and make artificial 'components' from tinned copper wire - simple 'U' links - and solder them into random places on the board. There is no need to make the usual track cuts, even if one of your 'components' has both ends connected to the same track!
Try both straight-through (being careful not to burn yourself while soldering them and holding them in place!) and captive, where you bend the wires over after inserting them.
Now practice removing your 'components' ...
