Ok, using SQLyog we are going to export each item one at a time, this wat it insures nothing gets lost. *Turns on monitor to Mangos server and opens SQLyog* OK! This could get messy because I don't know how the stand-alone works so bear with me.
You should have 2 different databases, one called mangos(or mangos_bc) and another called realmd(or realmd_bc) Right click on the first one and select "Backup as SQL Dump", select where you want it to save everything to and it will create a dump file, now do the same with your realmd/realmd_bc file.
Now we have everything backed up. Install the mangos server and SQLyog on the new PC. Once mangos is installed use SQLyog to create two new databases (same name as the ones you exported from the old server) once this is done right click the first database (the mangos one) and select "Restore from SQL Dump) select the SQL file you backed-up that matches the database you just made, it should start importing it, do the same with the realmd/realmd_bc database as well.
This should be all you need to do *IN THEORY* I don't guarentee anything.