You'll want second hand parts of everything if you want to do it on a budget, in which case there is unlikely to be a specific cost we can tell you. You might find someone clearing their garage that has 4 perfect shocks for £10 the lot, or you might have to hunt around and buy 8 less than brilliant shocks for £20 each.......
You need to draw. First off draw a scale sketch of what you want in plan and side view, including the chassis tubes, wheel locations, steering column, engine, radiators, seats, pedals etc etc etc using your best guesses as to dimensions. Then gradually source the bits you want, either from a complete donor car or as individual parts, and put them into your drawing so that it gradually morphs from guess to reality. Along the way you will find things that can't be done due to lack of space, and will have to relocate things, but that's the joy of a project like this.
Once you have a reasonable idea of what is going where, and how each bit is joined together (i.e. try to consider coolant pipe routing, or throttle cable positions, or how the handbrake is going to be activated (if you want a handbrake), and what sort of wiring you'll need) then you can finalise the design, and start cutting/welding metal to form the chassis on to which everything hangs.
Then you start bolting stuff on, and at this point is where it becomes difficult. If you thought the first bit was hard, you ain't seen nothing yet!!!
Everything you fit won't. You will have to change stuff. Modify stuff. Rebuild stuff. Reweld parts of the chassis to accept the changes. Even once you have a rolling chassis, you're only about 10% of the way there. You need to think of EVERYTHING at EVERY stage, bearing in mind that a lot will change after every stage. Write everything down - part numbers, dimensions, drawings of modifications, costs. Make sure that you have a drawing that shows the current state of the design at all times (i.e. if you change a chassis tube, annotate the drawing)...
Have fun! Sounds like a wonderful summer project. You'll need books (no recommendations, just read up on what you need to know at the time - wiring, engines, stress calculations in trellis frames etc), you'll need tools and you'll need to not stop thinking about it until it's finished.
One of my life ambitions is to design and build a road going sports car myself (well, as much of it myself as is feasible), but I'm not clever enough yet, I don't have enough time, don't have enough money etc etc etc. I think it's something I'm more likely to attempt a decade from now.