Yes you can do your website SEO yourself if you understand the basics and have enough time. There are plenty of FREE and paid tools available in the market today that will guide you to get started/enables diy SEO.
- Google Search Console (Free): This is one of the most powerful tool for SEO. You can get the search analytics data through its API and you will get a good amount of insights and actionable items to work on. It will also give you lot of other information pertaining to errors.
- Google Analytics (Free): Using this you can look into the user engagement numbers and try to improve them.
- Google Trends (Free): Don't miss out on the trending things and Google trends can help you a lot.
- Google News (Free): Like Google trends, you will get to know what is latest in your field and you can quickly cover it on your website.
- Google Keyword Planner: You can use Google Keyword Planner to get keyword search volume data. Though this tool is meant for Adwords usage and the search volumes are different from actual impressions that you would get even when you are ranking on the 1st position, you get a good understanding of the keywords that you can use.
- Page Speed Insights / GT Metrix / Pingdom / Chrome Developer Console / Lighthouse (Free / Paid): All used for checking what you can do to speed up the website.
Below are some of the paid ones
- Ahrefs (Paid): This tool is primarily used to get an understanding of the competition. Ahref's crawler is one of the best and it will let you export huge amount of data. You will get a good understanding of the competition keywords and backlinks through Ahrefs.
- Semrush (Paid): Semrush is primarily used to crawl websites. The kind of data it gives is quite insightful.
- SERPLAB (Paid): Keyword rankings can be monitored effectively via SERPLAB.
- MOZ (Paid): MOZ is useful to get the DA of a website. MOZ also has a very good user interface it will give you specific instructions on what you should do on your website.
- BrowserStack (Paid): You need to check how your website loads on different browsers. BrowserStack is quite effective in it.
- Hotjar (Paid): Is used for understanding the user behavior.
- Copyscape (Paid): Plagiarism is a big no if you want to do a serious SEO, Use Copyscape to find plagiarized content.
There are also many other tools available in the market. However, for your need, the above ones will suffice.