You won't necessarily have to run Wine. In fact it's best that you try not to and instead use native applications. Ubuntu is a good Distribution to start learning Linux as is Mint. However keep in mind that there are many other choices out there and when you get more experienced with Linux you may find them more to your liking.
You'll find that there is a bit to learn and that it won't be completely effortless but if you stick with it things become much easier over time.
http://ubuntuforums.org is the best place to go for help with Ubuntu or to read about it.