Ok, went to Scotland last summer as a twosome with my father-in-law and probably have too much info for you but....first, I would reconsider using a packager if you want to play TOC....You can get a guaranteed tee time that way. I can heartily recommend Perry Golf or The Old Course Experience...you will pay extra, but can you really live with yourself if you go all the way to St. Andrews and not play TOC?...if the answer is yes, or you will have the opportunity to visit many times then by all means take your chance with the ballot....but remember, the ballot is very mysterious and even the locals cannot give you the full skinny on how it works.
Second, in addition to the other suggestions, I really liked the St. Andrews Golf Hotel....a small old hotel just perfect for golfers, just 1-2 blocks from TOC with a young, interesting, welcoming proprietor and a great bartender named Roy.
Kingsbarns is a must. Carnoustie is not far away from St. Andrews..maybe 40 minutes. Crail is liked by many and only 10 minutes away from St. Andrews. As a change of pace, the Dukes course was interesting (not links golf however) and has carts...maybe Dad will need a break from walking for a round.
If you like seafood and don't mind an expensive meal, call ahead and get a reservation at the SEAFOOD RESTAURANT....right across the street from St. Andrews golf hotel on the water.
You can get on TOC by arriving at dark 30 and playing as singles...I can't think of a worse way to ruin a wonderful experience ...but many do it and swear by it.
Make sure you have a drink at the Dunvegan ....
What else would you like to know???
Bart