The main GA airfields in the immediate Miami area are Kendall/Tamiami (west of the city, just the other side of the tollway/turnpike - no further advice offered on how to get there!!!), Opa Locka (north of the city, just off the Palmetto Expressway south on Red Road), and North Perry (north of the city and north of the turnpike). Opa Locka is probably the closest to Miami beach and has a few training organisations IIRC, whilst Kendall/Tamiami and NOrth Perry are further but both probably offer more choice.
Andy