Another Airline Ticket Cost Question

Thing is, we won't know what's a good price for you to fly from LA, only people living there will be able to really tell you that. If we flew from here to LA, our price would be very different to yours anyway. You might need to ask on the American forum to get views from people who have done that route.

Google flights from London (say) to any of the above places and you will get an idea of the price range.

Agree with Bellart 100%, but a couple of other things to note:

• London, along with Frankfurt, Paris etc is one of the most popular european hubs, so there is often a premium attached to flights via LHR. Budget airlines run from several other european hubs, so consider looking for a transatlantic flight into somewhere else. Istanbul (home of Pegasus airline) would be one obvious option for SE Europe.
• Have you looked at prices into Belgrade? Air France are offering $950 at the moment for your timeframe.
  

www.skyscanner.com - and don't leave it any longer. Prices rise closer to your flight date. Look at the 'whole month' option to find the cheapest date near your projected leaving time.

'Skyscanner' will be your fare finder friend for your long-haul question.
As for the internal (Europe) part(s) of your trip, check out Ryanair, EasyJet and WizzAir.
Be aware, all the above 3 airlines charge for cabin baggage. Ryanair and EasyJet are rapidly doing away with seat allocation costs though, so no need to pay extra or going for speedy boarding either. Wizzair has a stricter cabin bag size weight/size limit than the other 2, so please do a bit of searching before going ahead.

Mothercourage: You beat me by 2 minutes :)

I plan to travel through southeastern Europe for three months (Sep-Nov) -- and figure I might save some money by first buying a trans-Atlantic ticket (Los Angeles to London) -- and then fly to SE Europe on one of Europe's budget airlines.

Not necessarily. A flight from LA to somewhere in South Eastern Europe could be cheaper or, if not cheaper, could worth the difference.

First question: what is a "good price" for a round trip fare between LAX and LHR? Searching online, I see mostly $1,300. Should I buy or should I wait another few months? I should note that I no longer qualify for student discounts.

You could find at the moment cheaper fares to destinations like, for example, Istanbul or Athens... Take a look at webs like Skyscanner or Kayak (or any other similar, Orbitz...) for fares on your travel dates.

Second question: Which are the popular airports in southeastern Europe with cheap fares from London? Sophia? Bucharest? Belgrade? What price range should I target for?

As I said above, I think you should consider firstly if going to London and booking a separate flight from London to South East Europe is the best option. Bear in mind the cheapest fares out of London are likely to be with low cost airlines and you should look carefully their terms and conditions (baggage...) to avoid (unpleasant) surprises on the ground and to know beforehand the real cost of your flight. Look also the departure airport of your flight out of London and the arrival airport of your flight from the USA, the time (and money) it costs to get from one to the other, whether you need to stay overnight in London... and don't forget if you miss your flight out of London due to a late arrival of your first one you'll have to look and pay for alternatives on the spot what won't be cheap...

Cheap destinations out of London? As a general rule, any served by airlines like Ryanair, Easyjet, Wizzair...

Ben2World:
Popular airports in South East Europe are, as you state, Sofia, Bucuresti etc. Please bear in mind though, neither of those two operate direct flights to or from the USA, so you would have to change to get to either of them. Also, the lo-cost (budget) airlines don't always fly from major airports and tend to use smaller alternative/close-by ones. And this could result ending up having to change planes and end up departing and arriving away from the major city you're heading to.
As mentioned by poster @#6, a flight into London could be a good option, but the likes of Ryanair and Wizz and EasyJet don't fly in or out of Heathrow, so you'd be looking at getting an inbound flight into Gatwick or Manchester and then a connection, or Heathrow and boarding a train, plane or bus to get to Gatwick, Luton or Stansted to get a forwarding flight. Wizzair, incidentally, doesn't have 'Any' flights to and from anywhere in and out of Manchester.
As far as getting to South East Europe directly from the USA, have a look at 'THY' (Turkish Airlines) fares. It flies direct to Istanbul and depending on dates, one-ways can be reasonably priced. I would suggest booking one way and getting your return flight from somewhere else, to save having to go all the way back to Turkey again. Pehaps look at options from Frankfurt and Schiphol (Amsetrdam), 2 of the world's busiest hubs and many flights to LAX.

Ah, Lenny - you have to get up very early to beat me. I'll swear I'm getting younger every year .......

Thanks, everyone! Good leads to follow up on...