The first thing I would caution you is that virtually every programmer has had the same thought some time in his career. What every senior person knows is that if one were to start over and re-build a program from scratch it is always better the second time around. You are not unique.
Then there is the parable "The grass is always greener on the other side of the fence". Couple this with the thought "How hard could it be?" and it is extremely tempting to want to go out on one's own.
You need to temper this enthusiasm (which I applaud by the way) with the realization that most programmers tend to be optimists and that is why they tend to underestimate the work involved. There is also a lot more to a successful business (like legal, accounting, customer support) than just a website. Once again it is tempting to think "How hard can that be?", but be careful to not underestimate their importance. Whatever you do, go into it with your eyes open.
As to the ethics, the most professional way to prevent an ethics dilemma is full disclosure. Your company hired you to work on their system, and part of that is the expectation that if you have ideas how to make it better you will share them with the company. If you hired a programmer, wouldn't you expect the same from him? Now let's say you have the idea "Our website is built with old technology, it would be much better in HTML5 and working on tablets." Put that idea in writing and send it to your boss. Chances are 1) He already knows HTML5 is the way to go, but is too concerned with other things. and 2) Your idea will promptly be forgotten.
If after sharing your idea they do nothing (which is the most likely outcome) and you still want to go do it, go ahead, you have fulfilled your ethical and moral duty to your old company. If they were too lazy or stupid to act on your idea it is now their own fault.
On the off chance that they actually take your idea and run with it (highly unlikely), then ask for a big fat raise since you just helped them make lots of money. At a minimum you will now have it on your resume that you were the one responsible for the profits (remember you submitted the idea in writing) which should help you secure a job with another company or a partnership in a funded start-up.
As to the advisability of actually quitting your job and starting a company once you have everything square with your present company, that depends on many factors. Not everyone is suited by temperament to be an entrepreneur. Further can you afford to live without a salary while building your company? Being successful takes a lot more than simply an idea.
If you do leave, it may be in your best interest to fully disclose what you are doing. Most people shy away from this and try to hide things, which often leads to lawsuits. A classic example of where disclosure worked out well is Bob Gore (not Al Gore the politician) who was working for DuPont. Wile working for DuPont he figured out how to extrude Teflon. He went to his bosses and suggested this could be a great oppurtunity. DuPont told him they were not interested so he asked them if they minded if he quit and started a company to do this. They told him that was fine, so he did. The product is GorTex and he has done quite well.