As many have already mentioned, building a jet engine from scratch isn't exactly easy or cheap. In your position I would get a battery, electric motors, propeller, microcontroller and a 3D printer, then figure it out from there.
I know it's probably a bit off from what you wanted, but it's a "cost efficient" way to actually built something. Once you have something flying you can upgrade from there...
I am currently in the process of building my own drone from scratch this way, designing the drone and programming the remote control, flight controller completely myself... it is/was a lot harder than I expected. So, what I am trying to say is: don't be afraid to use of the shelf components and don't make things harder than they have to be. Get started, then get creative.
Just my experience with this.
A New Zealand handyman with a passion for jet engines says he is building a cruise missile in his backyard using parts and technology freely available over the Internet.
To build a jet engine from scratch you would need a $50000 machine tool shop and several years of prior experience.
Alternatively, if you really do want to build a "jet engine" from scratch, consider building a pulse jet engine [2]. You might be able build one with parts from a home improvement store over a long weekend. It won't be a great engine, but it might be possible.
https://www.youtube.com/results?search_query=build+a+mini+je...
- Is your government okay (both in theory and practice) with you building such a thing?
- How high-performance do you want it to be?
- What's your budget, both time and money?
A gas turbine? Solid fuel (like a bottle rocket?) Liquid fuel?
> I know a lot of physics is involved in building one.
+ your comment here makes me think you have a lot of research to do.
You should focus on getting an off the shelf propeller based remote controlled plane if your budget is also only $300 and not $3000+
See something like this
https://www.horizonhobby.com/product/aermacchi-mb-339-arf-wi...
And
http://modelaircraftcompany.com/newshop/en/micro-jet-turbine...
If you want to build a jet engine then go for it, but if you really just want to build planes then you can buy electric/piston/jet/rocket engines for much less than it would cost to build your own.