As someone who also loves to play with the exploration of the board, I do this simple rule to allow for lots of extra fun:
Always, ALWAYS draw an Encounter card for the Investigator, no matter if they've used the spot's special ability or not--and even if they closed the Gate during that turn too. That way they always get to do or see something unless they're parked in the street. Moreover, at the start of the game, after you've set everyone up with their gear but before you do anything else (including the Mythos phase) have everyone draw an Encounter card for their Investigator's location. Heck! Pull off drawing a Mythos card for the opening and wait a full round! It lets everyone get at least two whole Encounters before they have to start running around like chickens with their heads cut off.
As for playing with just the expansion, like the Professor said, you really can't play with just an expansion since a lot of stuff is based around Arkham (especially all the other worlds--you'd run out of Gates really quick if you, say, just used Dunwich's gates and then BAM! Instant AO!).
Ursula is a character designed to be used for exploring Encounters, so I would recommend following her lead on everything except her "Adventurer" ability. If you want people to have even more fun, let them chose their starting location, even in the Expansion board. It makes the game a smidgen bit easier at the start BUT it will allow for people to experience all manner of Encounters.
In the time I started using these few basic rules, I realized just how much you miss playing only to close Gates.