Hi, Emily,
You should have no trouble at all taking the train from Edinburgh to Aberdeen, Aberdeen to Elgin, and Elgin to Inverness.
There are several private hire companies who can pick you up at your lodging in Elgin, and take you around to the various distilleries. You can find them on the internet. There are also companies who can pick you up in Inverness and take you to distilleries near there. Perhaps someone on this forum has used such a firm in the past, and can make recommendations.
Taxi service is widely available in Aberdeen. There is a taxi rank at the train station, another at the top of Marischal Street (near the Castlegate), and one at Back Wynd, next to St Nicholas Cemetery. There is also one farther along Union Street, just off of Chapel Street. There are also taxi ranks in Elgin and Inverness near the train stations (Inverness is just outside the station door). You may want to call ahead for a taxi in Elgin - there aren't as many taxi services.
Uber has just recently become available in Aberdeen; however, I think that you probably would need an application for that, and I'm not sure whether a U.S. application would work in Scotland. In addition, the taxi drivers are not too happy with the new Uber service, so it's probably best to stick with the taxis to avoid ruffling any feathers!
If you're looking for good pubs in Aberdeen, I can recommend the Prince of Wales on St Nicholas Lane (just off Union Street, near the St Nicholas mall), and Ma Cameron's, on Little Belmont Street. Both are very friendly pubs, with decent prices and very good pub grub.
Best wishes, and safe travels!
Mike (Auchterless)
p.s.: In Elgin, I can heartily recommend the West End Guest House on High Street. Ross and Blair are wonderful hosts, the rooms are very comfortable, and the breakfasts are delicious. Plus it's a very short walk in to town.
