+1 to Gogg’s recommendation (The Ship & Mitre, Dale St). It is an easy walk from the station, if you are using the train to get in, and they serve food. However, it will be busy and it is also a bit of a walk from other eating / drinking areas. Having said that, you could visit Thomas Rigby’s further up towards the river. If you are still out after midnight, it is a short shuffle to the Matthew Street area which is a pedestrian area full of bars and clubs.
The alternative is to head to the Philharmonic. Grab pizza at the Quarter. Then head down the hill, potentially via The Fly in the Loaf (bit expensive), the Pilgrim (perhaps a bit studenty), the Roscoe Head (small) ending up at the Wood St / Fleet St / Slater St area which is rammed full of bars. This area stays open til the wee small hours and most places won’t care if you are in jeans. The Shipping Forecast is ok for beer and there is usually somewhere to stand if not sit. Expect all pubs to be very busy, it’s a Friday night.
I would avoid the Lime Street row of pubs, if you come in by train.