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.

I forgot to mention:

There is a smallish fresh market in Ta' Qali close to American Embassy. Produce is fresh, mostly local and in season and prices good. You need a bus to there though.

The main market Sundays in Marsaxlokk is a fantastic day out for its sounds, smells colours, people and variety.
There are a lot of fruit and veg stalls, upwards of 25. Prices and products vary greatly from one to next. If you know your prices and take time to look around, you will find a lot of good produce at best prices in Malta, not all from one stall or the first one you see!

For fish, the choice and freshness is the best you can find, but I don't find prices there good value for money. Also in hot weather fish can go off before you get home because it's two bus rides away taking at least 1.5-2 hours.
Marsaxlokk would be great for buying fish, fresh fruit & veg and other things if transport wasn't an issue.

On a positive note there are a lot of fish restaurants serving generous-size fresh fish dishes at very reasonable prices.