Can everyone see my Tweets?
The default setting on Twitter is that "Tweets" are publically available to anyone regardless of whether they have a Twitter account or not. Often this is of great benefit, if you are sharing information. Twitter is a great way of researching and discovering new information. There's an article here that explains How to Search on Twitter. So the fact that tweets can be viewed publically is great when you are using Twitter in this way.
However, as an alternative, Twitter offers a "protected tweet" setting where you can "approve" who follows you, your tweets are only visible to those who you have approved to follow you and your tweets can't be re-tweeted by another user that you haven't approved.
Whether or not you choose to have your Tweets public or private is entirely up to you. Personally, I have mine set to public to allow people to view them and choose whether or not to follow me.
You can find a more detailed explanation of public and protected tweets on About Public and Protected Tweets article.
Hope this helps