Local vegetables, Meat, Fish - St. Julian's, or close by

There are no markets as such, not in the St Julians/Sliema area at least. I think there is a market in Valletta, but not sure of the day. There is a big fish market on Sundays in Marsaxlokk, about 1 hour bus ride from Sliema.

I personally dont like the little street food vendors, the quality is not that good. Tower supermarket in Sliema has a huge section of imported fruit and veggies, fairly cheap.

For fish there is a small fish store in Sliema, search for Triq il-Kbira, it is a bit further up from Hole in the wall pub. They sell VERY good fresh fish, both local and imported. Local Maltese tuna is 15€/kg!

Seek and ye shalt find. Just go for a walk and take a look around. The islands absolutely abound with outlets.
Also veg and similar don't look as good from the street hawkers because they are not massed produced for supermarkets. so they will not look 'perfect'.

We were justs at a place called Azzopardi fisheries, it is about 3 minutes from the place we live in Gzira but it is also fairly close to Sliema. It had one of the biggest selection of fresh fish I have ever seen in my life, and I am a Swede so that tells you a lot highly recommended, and it was cheap as well. 2 kilos fresh local tuna for 22 euro.

Try Greens, Triq Ta' L'Ibrag near Pembroke P&R in St. Andrew. 50m from there there is also an Azzopardi shop for fish. The Spinola fishermen sell there catch of the day in the small shop in Triq Wied Ghomor (opposite of Ryan's Pub).
Likki

http://www.azzopardifisheries.net

Some of my recent catches there:

Fresh tuna steak 10.00 €/kg
Fresh lampuki       6.50 €/kg
Whole salmon       7.50 €/kg

They had many other to choose from.