Castle Street Townhouse, which is on Castle Street just off Dale Street & Water Street.
Then just up on Chavasse Park there's Loretta's Tavern & The Botanist - both nice little spots when the weather is good.
If you fancy little stroll over to the dock - both The Long Shot sports bar & The One O'Clock Gun are decent little spots for a drink. The Long Shot does some very decent bar food as well.
You shouldn't really have an issue with crowds of younger people as long as you stay clear of Concert Square, Slater Street, Seel Street & Bold Street later on. The places I've recommended are in probably what I would consider the nicer/tamer parts of the city centre (that aren't a bit 'out of the way' like Hope Street) and I live here so..