If you are new to this SEO thing as I see you are also very young, then you could maybe not take money as important now and do some practice to better learn actual SEO rather than trying to find the quickest way to spend the client's money.
There is not one best thing to do when it comes to SEO. You need to do all of them. Backlinks, create informative content as well as keyword targeted content, and also do social media…
I would rather do what is most likely to work and get positive results, rather than waste time and money learning new tools that will not help you actually learn SEO and that might even get negative results!
Here is what I would do in your situation:
I would focus on doing on-page SEO. Look at the competition and see what they are doing wrong on their website.
Get a semrush trial or maybe try ubbersuggest. Or you can also manually search in google and analyze the pages in the first positions.
The point is to check your competition ranking pages for the keyword you want to rank. A tool will also help you to check their backlink profile and authorithy score. Those are important metrics to guide your action but in your situation I would still focus on creating a better landing page than the competition.
See what they are doing wrong on their page in terms of On-Page SEO. Maybe they only have the keyword used 2-3 times or they don't have the keyword used in the header on the page.
Check the quantity and quality of content on that page that is ranking first. Maybe they don't have images and hyperlinks to external and internal links.
Maybe the images don't have a keyword in their filename and no image alt tags.
The point is to make a better page than the already ranking website for the keyword you want to rank for in terms of On-page SEO guidelines and best practices. And also visually to look nicer than the competition.
Educate yourself on On Page SEO as it's not too complicated. And apply on the real estate website as you learn.
I would also spend time manually submitting to local directories using the same details on every website (same address, phone number, business name).
Also, check if the client has a Google My Business profile. If it does, then tell him that reviews are highly important for SEO and to ask his previous clients to leave a review. If it doens’t have a GMB page then create it yourself or tell him to create it.
In my opinion the best you can do is to bring positive results, but not to rank him in first position quickly. I would still focus on creating a high quality page for the targeted keyword rather than focus on creating low quality spammy backlinks.
Maybe you can research and write yourself the content for the landing page (not a blog post, but a page as a landing page) and pay someone to rewrite profesionally for SEO.
And again, I would create manually submit the website in relevant web directories.