Hello community,

I just got BSG and the Pegasus Expac.

Any advice on learning the rules for this game and common issues or problems to look out for?

Thanks.

It seems way more complicated than it actually is.  On their turn each player moves then takes an action (you cannot move after taking the action) from a space on the board or a card in hand or a character ability.  Then you draw a Crisis card and resolve it, it will either be a skill check or the specified player will make a choice between two really bad options.  You then activate Cylon ships if applicable and move the jump track if applicable, then go on to the next player's turn; after a few turns of playing everyone should get it. 

Make sure you read the "frequently overlooked rules" section so you aren't stringing XOs together and such.
Keep in mind it is a very social game and should be played as a thematic team game, some people have reported that when they play they all fight over the titles (makes sense if you are playing Zarek I suppose) and then wonder why they lose.  Work together and trust no one until the Cylons are out, but don't doom humanity because of lack of trust, more than 50% of the players do actually want the Human team to win.
Don't XO the player after you if there are still Cylons in hiding; if that person is a Cylon, you just handed them 3 actions in a row to screw you with.

That's a pretty good primer, actually. Eventually these basic mechanics become automatic and the real conundrum is in making a sequence of high stake risk / reward calculations based on limited information. The extent to which you enjoy being in those positions and your talent at working through them will have a big hand in your eventual like or dislike of the game.