Best Tapas in Malaga
El Pimpi is an inexpensive restaurant in the center of the city (Calle Alcazabilla) where you can try many typical Spanish dishes and tapas.
My favorite tapas bars are Los Gatos and Casa Lola in center of town. Also try fried fish (very typical) at "Pepa y pepe".
Outside the center but not far away, very famous for fried fish is "El tintero", a snack bar, where the waiter will pass by you shouting what he offers and you are free to take what you want.
Best paella in Malaga
For the best sea food paella, head to one of the chiringuitos (beachside restaurants) found on all beaches of Malaga. My favorite is El Tintero alongside Playa del Dedo. If you are looking for paella in the town center, I would suggest menu del dia in restaurant Axarquia.