We do Mexican well, Italian not so much.
In the Gaslamp for Italian try Bice (a chain, but quite good), another decent Italian place is Il Postino in the North Park section of San Diego (about 2 miles from the Gaslamp).
Really depends on what type of Mexican you're looking for. For Alta Cocina, you're pretty much out of luck oddly.
For tacos, one of the best places to eat is Tacos El Paisa - seriously authentic Mexican street tacos cooked over real coals. Super cheap, my family of 3 can eat there for under $10. Located East of downtown in Sherman Heights/Barrio Logan. Some might call it a "rough" part of town but it is not - lots of families and a police station across the street.
We also do really good Mexican seafood tacos. Try the Seven Seas Food Truck (yes a food truck) in the South Park neighborhood in the Gala Foods parking lot. The Gobernador/Governer is not to be missed (shrimp with chilies and cheese). The smoked Marlin is out of this world.
Super Cocina in City Heights is considered one of the "best" places for homestyle Mexican cooking. This is guisado or steam tray dining. Walk in, ask for sample, order after you've tried everything.
If you kids are fans of Bizarre Food with Andrew Zimmerman, try Aqui es Texcoco - this is all about lamb; lamb tacos, lamb stew, lamb tongue, lamb brains, etc... and is seriously delish (well not the brains for me). About 15 minutes from downtown San Diego.
Located around the corner and also Zimmerman approved is
El Gallito Tortas Ahogadas - another food truck dedicated to Mexican sandwiches. Order it dunked. Do not order it spicy! - Seriously. Can peel the varnish off an aircraft carrier. But one of the most delish things you'll ever eat.
All places are cheap and as authentic as you get as they cater to our local Mexican population.
If you are looking for "authentic" I would do my best to avoid anything in Old Town which caters to the tourist trade and offers nothing memorable besides a lovely setting (which counts for something from time to time).
Safe travels!