Trip Ideas - Italy? Spain? Greece?

I don't see anything wrong with adding a short stay in Barcelona to a longer stay in Italy, if that's what you want to do. However, if you are planning longer term you may be better to sketch out 2 separate trips, as both countries are better served and more comfortable with 2 weeks each.
Barcelona (4 nts) >> Venice (3 nts) >> Florence (3 nts) >> Rome (3 nts)

You can keep more of Spain in your back pocket for a further 2 weeks in the future.
Madrid (4) > Granada (3) > Ronda (1) > Sevilla (3) > Cordoba (2)

I'd say fly into Rome and visit 4 days. Then go up to Florence for 4 days via train. It'd be nice to rent a car in Florence and ramble through the Tuscan countryside.
And it's an easy train ride from Florence into Venice for 2-3 days.

From Venice, you can fly to Barcelona very inexpensively on Vueling Airlines.
Barcelona has a good international airport to fly home from.

Your time means, presumably 13 days actually in Europe, the first day of which at least will be getting over jetlag. Every time you move on, you will lose at least half a day in transit, so try to move on as little as possible to maximise your time.

There are many highlights in both Spain and Italy, so assume that you will be back and just focus on one country. The itinerary depends on where you fly into. Check out rome2rio.com to plan your route. Personally, I think that the food is better in Spain. Italian food is better in London than in Italy from my experience!

I would visit Madrid, Barcelona, Granada to see the Alhambra and Seville. There are many places that you can cover in day trips including the coast, Cadiz etc. Although some of these cities can be expensive, they are not in the same league as Rome and Venice. September should still mean good weather but without the hoards of school children.

Greece or Spain, you'll need to fly to or from Italy. As long as you don't try to string together 1-2 night stays, it won't be exhausting. You just have to consider carefully what your must-visit places are. A lot depends on what your options are for "good" flights (few, if any connections) to/from LAX flying open-jaw. If it works out, Barcelona would be my recommendation for several nights after Italy (so you get there without jetlag), and starting in Venice, the best place to recover from the flight while just soakin' up the atmosphere.

Hopefully, someone who travels frequently from LAX will pop up with some suggestions.

kn,

It should be easily possible to work out a fairly relaxed Itinerary for Italy and Spain to cover a two week time frame. However you'd need to limit the number of stops in each country to a reasonable level. A week in each country will only provide a "sampler". The other factor is that there are reasonably good flight options between Italy and Spain.

Greece is a bit of an "outlier" given it's distance from the other locations, and also because getting around there is a bit more challenging. Of course, as always using open-jaw flights would be the best idea. If you only have two weeks, also keep in mind that will only provide 12-days of actual touring once your flight days are deducted. From the west coast, you'll arrive in Europe the day after you leave home, so that reduces your trip by one day right from the start.

Before suggesting an Itinerary or commenting on "airline suggestions", it would really help to have an idea of which places you want to see in each country. Also, is this your first trip to Europe?