It looks like a lot of the Lindsay heritage is in Fife and the and the Central Belt. Have you found this link? The Kingdom of Fife is an interesting part of Scotland. There are lovely little fishing villages on the coast of the Firth of Forth. There is a good palace--Falklands. It has a very early tennis court! Of course there is St. Andrew's which in addition to the golf course has a great castle ruin. There is a mine that you go down into that relates to the reformation wars.
One of my very favorite castles is Glamis. This is the home the Queen Mum who died a few years ago. There are wonderful stories and it starts with a very old Keep.
To see Fife it would be best if you had a car. So, think about that in terms of your budget. You probably get to some places through buses, but it will restrict your schedule. You don't need it for the whole trip so try to figure out the schedule an only get a car for the time you need it.
You should explore Stirling Castle. The Castle is wonderful. If you are really wild about castles think about Doune. It is the castle that was featured in Monty Python's Search for the Holy Grail and more recently in the Starz Outlander series. I enjoyed my visit.
As Monte suggests you should pick up a guide book. I would recommend something other than Rick Steves as his is a bit focused for my taste and experience in Scotland. I really like the Footprint Guides.
Once you've looked at the guide books a bit, come back with more questions. :)