Search Engine Optimisation should be considered as part of a long-term strategy. Every website owner wants to be at the top of Google and Bing, so the search engines ensure that only the most credible, relevant and authoritative websites get there. Its not as simple as adding content to your site, creating a social media profile and setting up a Google My Business account. SEO is a complex web of hundreds of ranking factors that need to be covered. The more you have covered the less volatile your rankings will be when the search engines update their algorithms.
You should almost certainly see an improvement in your rankings within the first month, however, the amount of time it will take you to get to the first three rankings in Google will depend on a number of things. First, how much SEO you want to commit to each month, how well optimised your site already is, how strong your competition is, how functional and enticing your website is and how well your website is built.
I usually recommend to clients to see out the first three months before determining whether it is working for you or not but you should definitely be seeing improvements month on month in the first three months.
If you are looking for faster and more immediate results, I recommend starting with Google Ads.