there is a quite similar discussion going on here
Grenzübergang Russland Mongolei? (Länder)

Currently there are 3 borders open to foreigners:

Tashanta/Tsaagannuur in the far west - quite appropriate if you want to visit the Altai mountains and want to see a lot of Mongolia (UB is about 1700km away)

Altanbulag in the centre of Mongolia - just 300km away from UB, some major sights of Mongolia are quite easy accessible from there

Erencav in the east - don't know much about it. The scenery is flat grasslands, the only major town around is Choibalsan (but nothing quite worth visting)