I have been container gardening for about 6 years now. Have had some good years and some not so great years.
I definitely recommend buying starter plants, too late for seeds. I use large plastic pots. You can find them pretty much everywhere that has a garden center. I highly recommend miracle grow potting soil. It does the best job IMO.
It can be pricey the first year. Big containers can run $30 or do a piece. Potting soil and plants, depending on how many you grow, it can all add up fast. It is an investment of sorts if you want nice looking pots, but you never have to buy the pots again if you buy the plastic ones. Oh and don't forget to drill drainage holes in the bottom of your pots!!! That is usually not done but they have them marked of you look inside the pot.
I currently have 3 different varieties of peppers, two different tomatoes, and cucumbers. Cucumbers are climbers but still do very well in pots. They just spill out and over so make sure that pot has plenty of room on all sides. Make sure you buy at least two or three plants of the same variety, to plant in the same pot. It will make your success more likely.
I have done eggplant in the past. It was great except all of it matured at once and I had to come up with seven different eggplant recipes. :) also, once you pick the eggplant (unlike tomatoes) that was it, nothing grew back. The growing season was long for that I& I started it from seeds. I also did romaine lettuce. Once you cut it, that's it. Peppers do really great in containers, and more keep coming as you pick. Tomatoes are the same. I do one batch(3plants) of cherry tomatoes and a batch of large tomatoes. Those I hang on the garage w a topsy turvy.
The biggest thing is to make sure they get the sunlight and the water that they need. Also, when using the really big containers, you can fill the bottom of the containers with plastic water bottles (learned that on HGTV) the roots won't go all the way down the really big pots and it will make it lighter so you can move it if you need to.
If more people knew how easy it was to do, I think that they would do it. Oh and as a side note..., I have never, in my life, been able to keep a house plant alive, and still can't!
Good luck & happy planting.