As a former resident of Northallerton perhaps i could be of some use:
1) As SWT mentioned there is the Joe Cornish Gallery in Zetland street about 15 minutes for the station
2) Bettys in the High Street is very popular if your budget will stretch, I can recommend a fat rascal and a pot of tea
3) Couple of decent pubs. Recommendations are subjective but I like, because they do a decent pint of ale, the Tithe Bar on Friarage Street (more expensive but nice and no chavs - open form noon) and the Tickle Toby on the High Street. Less selective but a decent pint. The Station Hotel was closed for a refurb last time I was there. If it is open it is, surprisingly, next to the station!
4) The walk from the station is down South Parade and the High Street, both very pleasant roads if now busy with traffic. South Parade is tree lined and has fine Edwardian and Victorian Villas. The High Street is a much older street and is very wide with some nice buildings that are a little spoiled by modern shop fronts. A the bottom of South Parade is a Tesco if you need supplies
5) There is a market on a Wednesday and Saturday on the High Street
6) County Hall is right at the front of the station and is a nice building, as ids the court house round the corneron Race Course Lane ( sorry the race course went in the late 1800’s)
7) Further afield is the Wensleydale Railway at Leeming Bar. The Dales and District no 73. Bus goes from right outside of the station. It leaves at 09/39 past the hour and arrives at White Rose Hotel in Leeming Bar (opposite the station) at 24/54. Trains leave Leeming at 10:00/11:30/13:15/15:00 & 16:00. (The 15:00 is the last round trip) The round trip takes about 2hrs 45 minutes. The normal operation is a heritage DMU (101/117/110). During the summer they run two trains being the DMU and a diesel hauled train. They have a couple of 20’s a couple of 47’s a 14, a 37 and a 25 in minus a cab! (oh and an 03). It is a nice little run though some decent scenery. ( I doubt you will have time for more than a round trip at best but Leyburn is a really nice Dales town and worth a look)
8) Richmond Castle is really nice but is a bit of a journey (in fact you might be better form Darlington!) You get a fantastic view form the top of the Keep and Richmond is a really nice town. The old station at the bottom of the hill is a good place for a quick meal and a drink. The market square by the castle is also worth exploring. The Green Howards Museum is also in Richmond market sqaure For the bus you will need to walk down to the Buck Inn by the church at the far end of the High Street (20 minutes) and look for a Dales & District 54/55 ( the 54 may pass the station but I cant remember. They are not frequent!
The station has no gates, a small ticket selling building with a tea hut run by the Wensleydale Railway. Prices are decent. A shelter on each platform with a posh bus shelter on the southbound platform. That is it. If you want a fag walk you will have to go off the station on the south bound platform but northbound walk to the very north end of the platform and down three steps into the car park
TBH there is isn’t much to do. Depending how long you are there have a wander down the high street, have a coffee, find a pub, read you book and get back on the train!