Where to stay between Edinburgh & Glasgow

Being perfectly honest, as someone who is from west central Scotland, I'm struggling to think of places I'd really recommend with that criteria. My instinct would be to pick one of the two cities and go with that. Trains between Glasgow and Edinburgh are fast and frequent. You've got links golf on the west coast in Ayrshire [edit: just west of Glasgow] and St Andrews is north of Edinburgh.

The obvious suggestion would be Falkirk, sticking to your question.

I'm not from from west central Scotland, but I've been to Scotland many times. And I really can't think of a place to recommend. You might want to consider Stirling though, even if it's a bit to the north.

Stirling is a good call. That's a bit prettier than Falkirk. I'd still personally prefer to be in Glasgow or Edinburgh, even in the leafy suburbs of one of the two. Glasgow's better for the seeing the countryside aspect. 20 or 30 minutes north of Glasgow, you're into some very attractive landscapes as soon as you pass Balloch, and it just keeps on getting better the further north you go.

Most of the towns that that fall into the category of "between Glasgow and Edinburgh" are a bit too gritty / boring / depressing [delete as applicable] to recommend to visitors taking a vacation. North Lanarkshire and West Lothian suffered quite badly when the steel working and coal mining industries ended. In more attractive places in these regions you might struggle for public transport.

If I was plumping for somewhere I would suggest Linlithgow. And, as you have a car, I might suggest Biggar. It's a bit far south, but not so far as to be daft.
But really I think I would suggest one of the urban villages on the West side of Edinburgh, or maybe Stirling.
For Glasgow drive to Abington services from Biggar for the X74 bus hourly. For Edinburgh the 100/101/102 bus is hourly, from memory.
Historic Linlithgow is on the main train line between the cities.

Might Dunfermline fit the bill? Access to Glasgow and Edinburgh via public transport is pretty straightforward. I'm guessing you're going to need to use the car for getting to the golf courses.

Dunfermline's a pretty good shout. It's a historic little town in the centre and there's a fun connection with the US in the form of Andrew Carnegie. Friends of mine lived in a village called Oakley, just near there, and we'd be in Dunfermline quite a lot.

My knowledge of public transport in Scotland is quite lacking. I was a bit of an old car hoarder when I had the space to do it in Scotland and I'd drive everywhere. I'd associate Dunfermline with easy access to Edinburgh more, being from the west myself. Falkirk was the one place I know for sure has fast, modern direct trains to both Glasgow and Edinburgh. [edit: I too was working on the premise that the golfers would use the car]

I was just checking the Biggar bus times, from that another idea sprang up, which should have been obvious- Lanark.
It's closer to Glasgow, on a railway line. A Royal Burgh since the 1100's, known for its vennels (closes or alleyways), the Loch, two golf courses, Covenanter graves and of course the World Heritage Site of New Lanark- an outstanding planned mill village from the late 1700's.

Lanark and Biggar are cracking places off the beaten path for the more adventurous visitor to see. Lanark is the only place I've eaten that famous Scottish delicacy, the deep fried Mars Bar.