Im going to get some heat from DJs about these methods, but here goes. Its sort of the lazy way.
First thing to learn is to match snares... When the snares sync up, 99% of the time you have both the pitches (speeds) correct. Do that until you can do it well.. All you need to do is simply grab a snare from the record and drop it at same time the snare of the other song. You might not be able to drop it perfect and its not necessary unless its way off. Here is what some DJs might disagree with. Play with speeds to match it up and if you feel the record your mixing in is too fast, slow it down by rubbing the platter to slow it down. If the snare is ahead raise the pitch until its matched then slow it down a bit (how much you slow down is where experience comes in). Wait like 10 seconds between each adjust to see if it needs further adjusting.
Song playing.. Boom Boom Tah Boom Boom Tah Boom BOom Tah
Song mixing in. Boom Boom Tah Boom Boom Tah Boom Boom Tah <--snare is too fast so you slow it down so you rub the platter
Boom Boom Tah Boom Boom Tah Boom Boom Tah <---snare is too slow so you increase the pitch
Boom Boom Tah Boom Boom Tah Boom Boom Tah <--Snare matches up
Soft of obvious, but remember, when your slownig down with the platter and it matches, you need to also adjust the pitch slower. If your increasing the pitch because its too slow, once you have a match start slow it down. Think of a cop catching up to a runaway.. He has to go 100 mph to catch up to someone going 80. But once he catches up he has to slow down to 80 or he will pass him.
After you learn to match the snares you need to learn bars. Find the first bar of the song your mixing in, grab it. Wait for the first bar of the song thats playing. When it comes in drop the first bar youre grabbing. The snares should match closely. Then adjust like how you adjusted the snares.
Your not really suppose to having two songs playing while the chorus is in, of course there lots of exeptions. What you do is drop it an instrumental so you don't have to time the breaks then mix out of the first song. Then drop in the vocals.
There is way more it then that. Im just lazy..