I have local knowledge as Earls Barton is just down the road from me and where I go to a family farm to get my fruit and veg at their small shop.
While you are in Earls Barton have a look around the Saxon church on the hill which goes back to Viking days. There is an interesting potted history at https://en.wikipedia.org/wiki/AllSaints'Church,EarlsBarton What that history ignores is the excellent ring of bells in the belfry, 8 bells in a full circle, 3 of which date to 1720, and 1 dates to 1775. The tower was last overhauled in 1934 by the famous John Taylor & Co of Loughborough. They are sweet.
Earls Barton is pretty much half way between Northampton and Wellingborough stations.
Advantages of Wellingborough station - slightly closer to Earls Barton by a choice of good roads, served by yellow taxis and minicabs. Train service twice an hour each way. EMR from London St Pancras International (upstairs in the northwest corner of the station) to Wellingborough on electric trains with a destination of Corby. Except a very few at peak times no other passenger trains stop at Wellingborough.
Disadvantages of Wellingborough station - the Down platforms used by trains from London have a sharp bend in in them which means that leaving the train involves a very large gap as well as a drop to the platform. Could be tricksy with kids and luggage and guitar - but people do it every day. When I travel with my wife she can't leave the train without (pre-booked) assistance. The EMR line was never part of Network Southeast so never benefited from lower fares. Fares are always significantly higher than on the LNWR to Northampton. Buses don't regularly serve the station and while there are yellow cabs as well as minicabs they don't always hang out on the cab rank and most people will call for a cab as they pass through Luton or Bedford - that's what I'd advise you to do.
Advantages of Northampton Station (my depot for a number of years) - a large station with modern facilities and a busy taxi rank. Taxis and minicabs can be called, but usually there are cabs on the rank. Once you get out of Northampton's local streets the usually (except at rush hour) fast A45 dual carriageway goes directly to Wellingborough via the south side of Earls Barton. Fares from London Euston to Northampton are more reasonable and Advance fares can be inexpensive.
Disadvantages of Northampton Station - traffic is usually fairly heavy in the town, the X4 (mentioned above by Stuart) at the bus station is inconvenient as the bus station is up a hill and not the nicest bus station in the country despite just being opened a few years ago. The X4 isn't the only bus linking Northampton and Earls Barton, the X46 and X47do too. All are express buses, serving local stops on some parts of the route, and non-stop on other parts. The X4 is more express from Northampton and the other two are local in Northampton, both on similar routes. Taxis will likely be more expensive to Earls Barton than from Wellingborough because Northampton is a few more miles.
One other consideration is that there are virtually no hotels in Wellingborough. There is a Premier Inn which is not on the bus line and a Travelodge also not on the bus line, and several very very budget hotels I would never recommend. The Hind Hotel is in the centre of Wellingborough near the Wellingborough bus station, not inexpensive, noisy, and in a very old building.
Northampton has a number of chain hotels - a Mercure (in what used to be the Park Inn), an Ibis between the station and bus station, gets low ratings, a Premier Inn east of the centre, on a bus route, and a low rated Travelodge. The town is a bit rowdy at weekends.
My brother stays at a nice B&B north of Earls Barton but really needs a car.