Get a piece of blank copper clad board. (You'll probably have to buy a larger piece than you actually need) and while you're there get some etching fluid and an etch resist pen.
Work out where you are going to put the LEDs and (any resistors required) on the board. Work out where you want the wires to connect. (bear in mind that this is going to be the UNDERSIDE of the board)
Work out where all the copper tracks need to be to make the necessary connections. Cut the board to the size you need. Clean this up thoroughly. It wants to be scrupulously clean.
Now with that etch resist pen, draw those tracks onto the copper. (make sure it's sparkly clean first). Wherever you are going to have a component or wire attached draw a little circle with a hole in the middle. Leave this to stand for about half an hour to be sure that it is good and dry.
Find a plastic dish that you don't mind ruining. (Maybe from a frozen TV dinner). Put some of that etching solution into the dish. You don't need it very deep, just about half an inch will do the job.
Place the board that you've prepared into the dish of etching fluid. Since it's your first time it would be wise to have it face up. Be patient, this takes some time. Every so often you can rock it gently to dislodge any air bubbles on the surface of the board.
One all of the copper has been disolved (leaving just your tracks) remove it from the etching fluid and wash thouroughly.
With a piece of wire wool GENTLY rub off the etch resist ink that you applied earlier.
Drill the holes where you need them for your components. Poke the legs of the components through the holes from the other side and then solder them in place. (do the same with some jumper wires).
Make a coffee and have a break.