I would associate myself with Morland's advice above and add a bit of my own opinion.

Personally, I think Covent Garden is a bit overrated at this point. If you end up there and are looking for food, I'd actually suggest some adjacent restaurants that aren't inside Covent Garden itself and have great food. The market itself, especially around the holiday season, will be rammed with people, though there are still some decent shops if you're looking for specific things. What kind of souvenirs are you looking for? You may find it easier to do some shopping on Regent Street or even in some other (slightly) less touristy neighbourhoods like Sloane Square.

I love Borough Market. Much of that is because there's such a great diverse range of food and foodstuffs to be found and I don't usually go out to shop for things (I'm an internet shopping kind of guy).