We visited Western Ireland by bike in 2011 and are seeing Scotland by car this summer - 2 weeks each time. On bikes, we weren't traveling super fast, so didn't even try to see southern, eastern, or northern Ireland (or Northern Ireland) that trip. Buses with Bus Eireann (with an Irish setter on the side instead of a greyhound!) can get you around western Ireland (and take a ferry or pricier airplane to see Inishmore in the Aran Islands), but there are few trains in the West. With just 10 days, maybe you can see as much as possible in a region (rather than the whole country) this time, and hopefully can see other places on your next trip(s).
August is Festival Time in Edinburgh (several festivals all at once), so it's full of Scots and tourists, and lodging prices (if you can find a place) are at their highest. We're flying from the USA to London, then taking a Virgin Air flight to Edinburgh - cheaper and faster than the train. As noted by Ed, you may find Ryanair or other budget airlines can best get you from Ireland to Scotland. Once again, trains and buses are available in Scotland, but trains don't go everywhere, and with buses you're on their timetable and they don't go everywhere, either. On the plus side, if your up for driving on The Left, renting in Scotland appears to be cheaper than in many European countries where we've rented a car. On the minus side, the rental companies seem to have a minimum age (it varies with the company), so that may not be an option for you. You'll have a GREAT time, even if you don't see everything in both countries in your 10-day timeframe.