Algarve coast in March

I stayed in Lagos in February and enjoyed it. It's not as built up as areas further east but is big enough to have options for shopping, restaurants, etc. I don't recall seeing anything closed for the season but I wasn't really looking either.
Bear in mind the 'local' people you will likely meet will be British expats, as the Algarve is a very popular area for the British. But, you don't have to worry about needing to speak Portuguese.....
I really don't have any info on night life.

I was in the Algarve in February and hated it. We actually left early. Things were closed, the place we stayed was moldy. I also felt like it was a really weird area - lots of shopping and restaurants catering to British expats so it didn't have any sort of local feel to me at all.

You would be better sticking to the bigger places at that time of year. Tavira is less touristy than Lagos, but both would suit. Both are slightly hilly. Tavira has a better rail service than Lagos.

There will be plenty of retired Northern Europeans on the Algarve at that time of year, particularly French and British and an increasing number of Canadians.

Many older Portuguese people don't speak English - do you speak Portuguese?

Walking from Tavira is interesting around the salt pans. Take the train to Vila Real and get the small ferry over to Spain for lunch. Walking from Lagos would be more interesting heading west on the cliff tops if it's not too windy and you need a map as the paths are not marked.

February is not the best month to visit the Algarve as it's not likely to be beach weather and is likely to be cool and wet.

I have been to Cabanas near Tavira in March and found most places to be open and mixed weather. There were many nationalities there.

Hire a car, which is a bargain off season for maximum flexibility.

Every restaurant on the Algarve offers fish dishes.