You need some solution and a contact case to start off with. They make rub and no-rub solutions. I use Aquify by CibaVision, which is a no-rub solution, but I still prefer to rub my contacts just to ensure they are as clean as they can possibly be. I put a few drops of solution in my palm, and rub the lens from the inside to the outer edges for about 10 seconds, then flip the lens over and repeat on the other side. Then I rinse each side with solution for 5 seconds. After that I do the same with the other lens. Then drop then in my contact case with solution, to let them store until I'm ready to wear them again. When I wear them again, I always rinse them with solution first, just in case any debris in the contact case might have settled on my contacts. If you don't want to rub them, just get the no-rub solution, and rinse them for 5 seconds on each side, then drop them in the contact case with solution. Most contact solutions say anywhere from 4-6 hrs storing before wearing the contacts again. My contacts are made to be worn for 2 weeks, then thrown away.
To put them in, just put the contact on your index finger, and hold your upper and lower eyelids open, the contact will go onto your eye easily in most cases. It takes practice, but you will get better the more and more you do, and find what works best for you when putting them in. To take them out, you just want to use your index finger and thumb and gently squeeze the contact with your fingertips until it comes out.
You really do need to get an eye exam though, if you don't want to damage your eyes. Poor-fitting contacts can cause discomfort, eye infections, and have even caused blindness in some serious cases. Your pupils need to be measured by an eye doctor, so that you get the right size of contacts, and can avoid problems like that. Not everyone has the same pupil size, so there is not a "one size fits all" with contacts. A prescription is required to obtain contacts legally, because they are considered medical devices, regardless of whether or not you actually need them to see. I wear Freshlook colored contacts without vision correction, and I did go have an eye exam and contact fitting, so that I could get my prescription and know that I am using contacts safely (right size for me, proper care, etc). It's really worth it to have the exam and fitting...we only get one pair of eyes to last us our whole lives.
You can go to a Wal-Mart Vision Center like I did, and get a pretty reasonably-priced eye exam and contact fitting. I don't have vision insurance, and mine was only $91. That included the exam/fitting, one trial pair of colored contacts, a starter kit with solution and contact case, and my prescription so that I can order from anywhere I choose, online or at any eye doctor's office. The eye doctor also trained me on insertion/removal of contacts, and how to care for them properly, both of which are very important with contacts.
I don't know of anywhere that will do it for free, but for me the $91 I paid was a small price to pay so I could know I'm taking the best care of my eyes that I can.