Expert? LOL Not hardly (but thank you for your confidence in me). I have lived in David for two years and learned a lot, but I'm a very long way from knowing everything.
Your cost of living depends on your style of living. Our budget (two people) is around $1100/month for basics, rent (385), food, utilities, etc., not travel. If you live in Panama City your housing is going to be considerably more than anywhere else. Other areas with a lot of expats tend to be higher also like Boquete, Coronado, Pedasi, etc. I would say on your budget you should be in good shape anywhere but you can go through money anywhere you live if you eat out a lot, love to shop. etc.
There are no beaches near Panama City. Coronado and Gorgona areas have lots of expats and lovely (but expensive) houses on or near the beach. Chame is also on the water but I don't know anything about it. Azuero Peninsula is popular and known for the beaches - Pedasi and Las Tablas are the main towns, but you might take a look at Chitre also. It's quite a bit bigger. There are beaches and well known surfing areas on the west side of the Azuero too but they are very remote, and you have to drive quite a ways just to get to a supermarket. That wouldn't be very good for your social life either.
Moving west, Chiriqui has some beautiful areas like Las Lajas and Boca Chica, but again they are remote from supermarkets and things you need regularily. For some it is worth it for the beautiful areas, though it may not help your social life either. Puerto Amuelles is on the Costa Rica border and has beaches too. Right now it is a very depressed area but there is development going on, so it may have a lot of potential in the future.
I highly, highly recommend that you do not buy anything for at least 6 months, preferably a year or more! Be sure you and your mother are both happy in the location you choose. Sometimes people find that after a few months something is just not working and they need to try something else. Also, after you get to know the area and make local friends, you will know much more about true prices and maybe find places maybe not available to the general public.
It is wonderful that you speak Spanish! You can live anywhere you want and make friends with expats and locals alike. The Panamanian women are really beautiful as well as kind and sweet, so who knows? If you live in a town with people, and you are outgoing and friendly, you'll be surprised at how easy it is to make friends here.
If you are interested in Pedasi, check out this blog. http://indacampo.wordpress.com/ They have lived there long enough to know the area well. As her about a radio station, how to find houses, etc. If you are asking about radio stations as potential work, that is a whole other subject. Unless you have skills that they can't find within the country getting a job here is very hard. Most expats either work remotely or create a business if they need to work.
http://www.panamaforreal.com/ Check this site too. He reviews various areas including costs of living.
I have blabbered on quite enough! I hope it is of some help, and have a great time on your trip here