I couldn't find any mention of it in the Beginner rules, but can a player acquire an additional Talent Tree like in the core rules (except with just the trees in the Beginner Set), or are they limited to their initial choice? I suppose you could allow it with the appropriate XP cost, but would it cause any problems with the simpler Beginner Rules?

I would just convert the beginner character to a core character and go from there. There's no real point in using the beginner rules once you've run the module.

If you want to keep playing the Beginner Game characters (which is a great idea; they are really well-written characters), you're pretty much on your own. There is no way set out in the rules to "transition" a character from the Beginner Game to the Core Rules. There aren't as many skills, and the talent trees are conflated and truncated. But it's really not all that hard: just do yourselves a favor and re-spec the characters using the character creation rules, being faithful to your vision of the character. The career names are the same, so it'd be easy to just pick whichever specialization you feel best suits your idea of the character and go from there.

Then, once you're done with character creation, add in the XP you earned playing the Beginner Game to buy some shiny talents/skill ranks/new specializations. Easy peasy.

There's no way set out in the rules, but it's pretty easy. I used OggDude's character generator to do the migration, they were all within 10XP of what you could build normally. I would just take all the characters and convert using whatever XP is needed, then take the max XP required and make sure everybody has the same amount. Some characters might get an extra talent or skill rank, but it's hardly a problem.

So the Beginner Set is designed to have limited usage? It's not like Basic and Advanced D&D from back in the day were each one was a separate but viable system? Good to know...just a taste to hook people and get them to buy the hard stuff (i.e., the core book). :-) At least you get the dice, counters and maps too.

Well, it's a pretty good deal. There's a free PDF followup adventure that is about 4 times longer than what's in the box. But yeah, get the core book, if you enjoyed the game already you'll enjoy it even more.