how do i choose keywords for your website without a big budget?

Add YouTube and Amazon autocomplete to the free list if they're relevant to you - they surface phrasing google's suggestions don't. Then before writing anything, check whether the top results are really answering the question your keyword implies. Sometimes google has decided the searcher wants something slightly different, and you have to write for that.

If the site has any traffic at all, start in Search Console - the queries report shows what you already get impressions for, and pushing something from position 15 onto page one is the cheapest win going. After that it is free stuff: google autocomplete, the People Also Ask boxes, the related searches at the bottom. Keyword Planner is free with an ads account; the volumes come in wide ranges but they are fine for comparing one phrase against another. Go for longer, specific phrases where page one has forum threads or thin pages on it - that is the sign a small site can get in.