Watch the Guitar Player Wanted: Vocals A Plus! online guitar lesson by Karan Andrea from Guitar Player Wanted: Vocals A Plus
My hope for you is that you get a steady handle on the concepts of singing - how you produce tone, how and where it resonates in your head and body, how to work through your break, and how to develop a mixed voice that you are confident with. The results you get will depend mostly on how consistently you do the exercises, and how well you can apply the ideas and techniques to your singing. But these are the basics, and how far you take them is totally up to you and what you need to do.
From here, the ideal way to improve your singing is to find a good vocal coach because it is always best to have an outside observer to catch, correct and fine-tune your approach. You might have a Community Music School in your area, or there may be a music department at a local college or university. These are excellent sources for inexpensive lessons from skilled teachers. You may have to interview a few coaches before you find someone who will address your needs.
Most likely, you just want to hit the harmonies in a Beatles song without embarrassment, and you probably don't want to sound too polished. This doesn't mean that you can't take voice lessons. However, I encourage you to use this course as an ongoing resource - and I encourage you to go as far as you can because I know you will surprise yourself at how well you can really do.