Well, when you say studio shoot i assume you mean a shoot with a solid background (usually black, grey or white) and either hot lights (continuous light) or flashes.
bare minimum would be 1 light (i prefere flashes, some like continous light more) and some sort of background (usually paper, but can be pretty much anything).
The rest of the advice is the same as for any shoot, lightshapers that works for your style of shooting (softboxes, beautydishes, snoots, spots, ringlights, etc. ).
What can be an issue in a studio environment is to control the light, how it bounces around the studio, so some way to block light is usually very good to have (normally screens painted black on one side (to block) and white on the other (to reflect).
Now to keep cost down, i would suggest going for a medium-high end budget light with a good reflector. Avoid the cheapest china brands, and if you can get a used Profoto/Broncolor/Hensel/Bowens/Alienbee etc. its probably better than buying too cheap. There are tons of ways to do simple softboxes yourself and even improvise using thin white fabric acting as curtains to mimic window light.
Then everything also depends on what you will be using as your studio environment, if its the living room wall or a huge former industrial building
As i suspect it will be the former rather than the later, i'd say keep it simple. Use whatever backgrounds you have, and use fabrics and wallpapers on screens to easilly switch backgrounds.
And another tip when it comes to backgrounds. If you have the space and money to buy paper backgrounds (on rolls), once they get a bit murky and gritty (which they do
don't cut them down or throw them away. Instead you can paint on them and create a new background with a unique look. Here is one example where i took a white background and painted with a normal household mop dipped in coffee =)
So you can get creative with very little means. Now once you start working with paying clients, your image becomes fairly important aswell, and if you use homemade softboxes with duct-tape and cardboard boxes, the clients may think twice... unless you produce outstanding images ofcourse
Good luck!