When you are in main menu, open console with Shift + ` (it's above Tab button) and type:
/map <map_name>
<map_name> values can be seen in MBII folder, there are .pk3 files with different maps. For example, there's mb2_duel_training.pk3 there. If you want to play that map, you type in console:
/map mb2_duel_training
And hit enter.
For the first time you'll be thrown back to menu with error "This gametype isn't currently supported in Moviebattles 2", in this case just open console again, press arrow up (this will return the latest command) and hit enter again. You may also want to set:
g_MinMBPoints 80
Also note that there are no bots for MBII yet, so you'll be alone on that map.
Have fun.
As for servers, make sure that your filter (in top left) is set to Internet.