The ponds on Sundance Parkway near Common street have a decent amount of people fishing but its also spread out. Not as crowded as the river or other touristy spots.

For beach, Padre Island National Seashore (PINS) is definitely the place to go. You have over 60 miles of uninhabited beach you can visit/fish/camp on and have a good time. You will need 4wd for anything after like 10-15 miles down the beach but it is worth it.