Has it been proven that plants do better with hydroponics rather than the traditional method, soil?
That's a loaded question.
Google will provide all kinds of support documentation to that extent.
I will warn you however that, as with most things, there are exceptions. Not so much with plants but with care and attention given.
I used to grow traditionally, but my garden is nothing but shale. Good to build on, bad to grow in.
I used to add compost and fertilizer, but my crops were mediocre and eaten by bugs. Until I took up hydroponics Â± 6 yrs ago. Now I have healthy (unpesticided) crops at a fraction of the previous effort (and cost).
Initial cost can be heavy, but taken over the years in use it is far less than what I used to pay for compost and fertilizer.
Is there a way I could try just a few plants before going nuts with Hydro? Can I do just a few inexpensively? Can household items be used, or is a lot of it purchased if I want to try something small?
There are some very simple set-ups to start with. All you need is a bucket and some styrofoam.
Cut the styrofoam so that its fits loosely inside the bucket. Cut a hole in the centre of the foam to fit a netpot (or a well perforated plant pot)
Fill the pot with wood chips and let it soak in water. Fill the bucket with diluted nutrient (buy at nursery). Place the foam (float) on the nutrient and the pot in the hole. The whole thing is supposed to float on the nutrient with the pot sticking about 1/4 inch in the water.
Plant 3 - 4 seeds. When they come up, keep the best shoot to grow on (don't pull, just cut them of), and away you go.
You are going to have to do more search than to post here; there are thousands of ways to grow in hydro.
You can start by reading.
Here is a link that might be useful: Hydro Book
I read a US govt document saying tomato hot houses are producting 15 times the yield per unit area. Think about it - year round production, increased nutrients, CO2 saturation, no weed or pest competition. Closer plant row spacing. 15 might be pushing it but it makes reasonable sense to me.
Think it might have been http://cipm.ncsu.edu/cropTimelines/pdf/USgreenhousetomato.PDF