My site is not showing on any search engines, what can I do myself to change this?.
First of all, do not expect things to happen overnight, it can take weeks, even months to get your site high on the rankings.
As to what you can do you yourself, please read the answer to question 2

Do I really have to pay someone to get my site listed on Google? or can I do it myself ?
The answer is NO to the first, and Yes to the second. NO, you do not have to pay anyone to optimize your site, and YES you can definitely do it yourself. Everything we do to optimize and promote your site can be achieved by anyone, however it is very time consuming, and can cost a lot of money, money that is completely wasted if not done properly.
For anyone wanting to try themselves, I would recommend studying the source code of similar sites that are on the first page of google. However, this is a very basic approach, and even with the exact same wording as the nr 1 site, your site will probably not get listed.

Once my site is listed on the first page, will it stay there?
You will always have to put in some effort / work to keep your page higher in the rankings. The site must be up to date, otherwise you will start falling, site linking is important, make sure you link your site to "prime relevant" sites. So once your site reaches the top, it does not mean you can sit back and do nothing.