Teaching while doing a PhD
Most doctorate programs have some amount of teaching/ta/facilitator/etc as a requirement of the program. Some people try to do more, some try to do less, but unless you're working in a research lab (with its own endowment :-) expect to do some teaching.
Re 'good skill': if your goal is to find a tenure track position then yes this would be good experience. Assuming that you enjoy teaching, and who here doesn't grin, this would be a good opportunity to enhance and refine your teaching skills.
Re 'lectures on research': be careful here. Odds are that any teaching roles you would be offered at this point in your academic career would be pretty low level undergraduate type stuff. Depending on your area of research you might also run into some problems 'presenting' un-vetted data. There are always exceptions to the rule but I wouldn't expect for you to be teaching anything so specific that your research would come into play. YMMV but that has been my experience (since you asked).
Suggestion: Be careful with overloading your schedule. Teaching is one of those jobs that finds a way to embed itself in many aspects of your life that you didn't expect. For example: I expected to spend my middle years sitting on the porch reading books with my wife, not reading term papers. Oh well. While it is good to gain experience you don't want to spread yourself too thin and suffer an impact to your research (remember, the reason you're here in the first place :) or your personal life.
Concerning your first point, teaching is a major component of most academic jobs, and in fact is the main part unless you're a superstar, so you should at least try it to see if you like it. I thought I'd hate it until I tried it my senior year, and it was incredible and changed my life. I love to research and have published several papers, but teaching is the reason I want to be a professor. So try it out!
On another note, your PhD institution will probably be much more gentle and helpful if you have any difficulties, while a postdoc or other 'real job' will expect you to already know what you're doing. So you should try teaching now while you have a safety net.
Teaching (being a teaching assistant) happens to be a requirement of my PhD program, and there's an option to teach more than two semesters, with your PI's permission.
If you're interested in teaching while doing your PhD, I'd recommend applying to programs where teaching is required (I did, for this reason). If this isn't common in your field, you could try applying at universities known for undergraduate teaching: they are more likely to allow you to teach as part of your PhD.