Can I do search engine optimisation myself?
Yes and no. Search Engine Optimisation is a very complex area of web development and is an ongoing task rather than a one-off deal.
In order to improve your search engine rankings without having to contact a web developer, you can ask other businesses or organisations, which are in a similar business to yours, to place a link on their website which points to your site. This is an online "vote of confidence" and search engines take this seriously.
Try to get listed in business directories, industry directories, link collections and so on and try to get as many links as possible out there that point to your own website. You should also offer to place a link on your website too which points to the website you want to get listed on.