no, not a swim coach, just a flounderer
The main problem with surf training is that there isnt a part of the body that isnt used when you surf. So surfing specific training really means whole body training, for both power (muscle power and sprint power) and for (some) endurance. Which means specialisation is pretty hard.
For swimming, if you are just starting out, I highly recommend starting off with the 0 - 1650 (or 0 -1500 in the real world) program here
http://ruthkazez.com/swimming/ZeroTo1mile.htmlafter you can swim 1500, then that is probably enough distance/endurance for weekend warrior surfers. So then start looking at sprints as per the above (indeed, you could potentially start sprints once you can swim, say, 1000m) mixed up with longer swims and whatever keeps you interested (for example, I currently do 10 x 100m sprints then swim a longer interval (400 - 500m) then play around with whatever I feel like for another 500 - 1000m (meaning I might mix up 50m sprints, or 200m at 80% or swim 100m but sprint for 20m each lap etc)). But whatever works, so long as you are getting a good workout (puffing, feel tired at the end). Swimming can 'encourage' you to swim at a slow speed in a relaxing manner, because it feels good; but this is very inefficient for fitness improvement. You dont get better at running a 5km race by slowly walking 5km.
As a tip, a pull buoy can make things a bit easier when you are starting out and perhaps struggling a bit; but dont worry about fins or hand paddles etc. Goggles and pool buoy, and a water bottle on the pool deck, is all you need (oh, and swimwear I guess). I really like using a stop watch to prevent me getting lazy, any waterproof watch will do (or there are swim specific ones - I use a garmin; but it works the same as a $15 one off ebay)
What you will find when you surf is that your paddling technique uses 70-80% of the same muscles as swimming, and the other 20-30% will get painful and be what holds you back in the surf. As Oldie said, pulling exercises should also be used and this can compensate for some of the bits that swimming doesnt cover (pull ups are very good). This wont prevent something feeling sore, but you can adjust your workouts to hit those sore areas more (eg if your triceps get sore paddling, do tricep excercises like rows more than pull ups).
Secondary focus to pulling muscles I suggest should be core, then posterior chain (glutes, hamstrings), then pushing
muscles (chest, biceps). But if you are doing weights, really a full body workout with a slight emphasis on pulls ups and rows is the way to go
Rest, eat well, keep limber (yoga).
Its easy....

until your first surf is in overhead white water close outs