Wine making is very, very easy. If you can make beer, you can make wine. I would suggest buying a kit wine from Wine Expert or RJ Spagnols to get you started. It will teach you the basics quickly. You can use your same plastic fermenter and then you need a 6 gal carboy. Follow the instructions in the kit and you can't mess it up. You just need the juice, top it up to 6 gal. with water and pitch the yeast. Let it ferment, rack to a carboy, add some clarifying agents (which will be in the kit) and then bottle. 6 gal. makes 30 bottles, and a floor corker comes in handy. Your LHBS may rent them. Then you let the corks set up for 3 days, lay the bottles over and let them age as long as you like. I only like red wines, so I am letting the 60 bottles I recently made age for 6 months to a year.
After that, you can try to source your own grapes or concentrates and experiment with oak chips and other additives. If you stick to the kits, you will get good to great grapes. If you source them yourself, you never know what you might end up with. If you have a winery near you, ask them when they harvest and you might be able to go on crushing day and get fresh juice.
Good luck!