Jim,
Since Rich has the north covered heres some suggestions for west and east.
West - suggest you base yourself in Prestwick or Troon. Courses to play that you've not previously;
Glasgow Gailes (home course therefore felt obliged to put it top of the list !) - fairly adjacent to Western, just over the railway line, has maybe more of a heathland feel with plenty of heather and gorse. Regularly an Open Qualifier.
Barassie (Kilmarnock) - haven't played it in a while but similiar to Gailes. Mix of older (shorter) holes and new (mostly longer) holes with bags of character.
Irvine Bogside - like the two above, not immediately beside the sea but still qualifies as a links. Small greens and plenty of undulations from what I remember from my one visit.
Prestwick St Nicholas - no doubt I'm in a minority here but I prefer this to the Old Prestwick. Old fashioned links with typical out and in routing. Holes nearest the clubhouse, both going out and coming in, have bags of character with loads of humps and hollows, gorse etc. The course has a fair dash of quirk as well.
Dundonald - adjacent to Western and Glasgow Gailes, relatively modern Kyle Philips design. Compared to all of the above, this feels very American and certainly long depending on what tee you play from.
Troon Municipals - there are 3 courses, all of which play inland from the coastal railway line but apart from 3 holes on the Lochgreen course all are defintely what you would classify as links. The smallest is Fullarton which is a par 66. This is more of a beginners/old mans course (I wonder how many I've upset with that comment !) but still good fun. The other two I would recommend before it. The Darnley is the shorter of the two, but for my money has more character and is more fun with some good shortish par 4 's amongst the gorse. Lochgreen, which has the three holes which are parkland, is an Open Qualifier course and is where I believe Nicklaus first qualified for the Open. Bearing in mind that they are muni's they do tend to be scruffy round the edges but then In my humble opinion thats the way links should look. The greens though are generally to a very high standard and in past times some of the best I've played on.
Belleisle (Ayr) - if your looking for a break from links, Belleisle is a good James Braid designed Parkland. Also a muni.
Jim, theres plenty of other courses in Ayrshire but those are the obvious ones that spring to mind. Each has its own merits depending on whether you are a bunch of low handicappers looking to get tested or whether you are looking for some fun golf.
I'm sure you know all about the St Andrews area so I will stick to commenting on East Lothian.
Gullane - there are 3 courses, no's 1, 2 and 3. All three courses start on the edge of town and and then go over a hill (which I think is called Gullane Law, although I could be wrong) play over the otherside and then come back again. From the top of the hill you can see over to Fife. No 1 doesn't get much love on this site, possibly because I don't think it is good value compared to other courses. Certainly I would save your money and play 2 and 3 which are similiar in character although 1 is longer than 3 and has more bunkers than 2. Greens are consistently very good.
Luffness New - adjacent to Gullane. Not played it but reputedly very good.
Longniddry - another one on my wishlist which I've heard very good reports about.
Kilspindie - old fashioned golf, short with small greens. Bags of character as has the clubhouse.
Craigielaw - just over the wall from Kilspindie, new course which built at about the same time as Kingsbarns. Good fun with some undulating.
The Glen, North Berwick - undalting course, particualrly the 1st and 18th, with some interesting/quirky holes like short par 3 13th (?) on the beach.
North Berwick is the obvious place to stay and all of the above are within a 10/15 minute drive time of each other.
Niall