I live and work in Newcastle, as do a few others on here. It's not a bad night out at all. Very visitor-friendly. We get a lot of stag and hen parties travelling from up and down the country and there are plenty of students about as well as the locals.
There's not a great deal to the Bigg market, but it does have that grimy quality folk seem to find attractive. The Bee Hive is an old pub with the cheapest beer in town. Just mind how you go and don't be an arsehole and you'll have a laugh. as eshiers1 will attest they have strippers in City Vaults on matchdays. Up the road, The Gate <a class="postlink" href="http://www.thegatenewcastle.co.uk" onclick="window.open(this.href);return false;">www.thegatenewcastle.co.uk</a> is a lot like the Printworks, with Tiger Tiger and Lloyd's and all that. There's also Mood nightclub which gets a bit daft sometimes. Not bad for a 'normal' night out. The taxi queue outside here is a good bet for getting home as it's well organised by the dibble and has a burger van.
Down by Central Station, you'll find Revolution, Perdu, Wetherspoons and a new Lloyd's, The Mile Castle. There's also North (much like the one in Northern Qtr) and inside the station itself there's Centurion.
Speaking of Jesmond, you can get a Metro through to West Jesmond station for a couple of quid, takes about 5-10 mins. Jesmond is the sort of place you can hang around all day and not hear a Geordie accent. Full of students. There are however a number of decent upmarket bars and hotels round here, mainly along Osborne Road. Expect to pay top whack for your beer, though.