Visiting daughter in October while she studies in Galashiels

you have a few options, Hire a car and Stay in or near Galashiels (commonly know as Gala) and spend time visiting all the Borders area you could even drive down and take in part of Hardians wall, but places like Melrose ,Peebles and Jedburgh are very short drive away .A day trip to Edinburgh would be easy enough too.
You could stay in Edinburgh no need for a car and spend time there but it is very easy to visit Gala now as the new Borders railway has just opened today and there is a station in Gala.
can't really recommend hotesl as I live in Edinburgh so don't actually use hotels in this part of the country.

My first recommendation is hire a car. The Borders is not one of the areas where you can rely on public transport.

Towns and places to visit, there are a lot. Abbotsford, Sir Walter Scott's home, is in the Borders, there are the abbeys at Jedburgh, Melrose etc, just across the border in England is Alnwick and one to check the tide times for, Lindisfarne, all depending on your interests. You can check out Historic Scotland, English Heritage, the National Trust for Scotland and the National Trust.

But the main recommendation is a hire car. And one from the smaller end of the market.

i agree, you need to rent a car. it is relatively easy exiting the EDI airport with the rental car but be sure to get the absolute smallest car you can fit into. if you are from the UK the driving will be easy, if you are from the US you will appreciate a small car as the roads are narrow, they curve a lot, and you will be on the opposite side. you might visit this link, http://trafficscotland.org/map/index.aspx?type=26 to see what the traffic is like during the time you will be on the road around EDI. Hopefully Pam will check in, she always has helpful info on location of travel sites in Scotland!!