Lets see. You have a few options here. Firstly, you're always going to have a much easier time in spring. This is when plants naturally want to grow, with damp spring rains, longer light cycles, warmer temps, and higher humidity.
However, you can simulate all of that. I think the easiest thing for your start with would be to get a mini hydroponics kit, such as:
These take a lot of the guess work out of it and give you instructions to insure that you have a much better chance of success when you're just starting out. That's the most important thing. I see to many people try any hobby and want to see progress too soon, giving up when they fail. It's even worse when you put a lot of time, money, and effort into it and it completely fails. It's very discouraging and its happened to me many times. With a kit like this, you'll get a good feel of what it takes and you'll have the appropriate light and nutrients for the plants. They also have many different capsules.
You can simulate this other ways by setting up a small grow station. You can get a light bulb designed to give plants the optimum wave lengths, a seed starter tray, and a warming mat. These will set you on a good road to success.
Lastly, you can try to just grow something in a window from seed. It may work out. You may want to try increasing the temperature and humidity by making a humidity dome out of something like an old, clear 2-liter coke bottle. Keep in mind that many plants started this way will grow great, but when you try to remove the dome and the temp and humidity drop to the normal house levels, it may be shocked and die. You just need to keep an eye on it and try to harden it off. Sometimes putting holes in the humidity dome and adding more over a few weeks helps acclimate it to the conditions.
I'll leave you with a last few notes to help you succeed, based on personal experience. Light, nutrients, and water are the keys to growing any healthy plant. Research each plant you plan to grow heavily and eventually you'll build up a good mental database of plant care. Lastly, don't put your hopes on one plant. If you do and something happens, it's heartbreaking. However, if you have a dozen plants going and 5 die, it's just a few and you still have the majority. I'd rather get to many going well and toss them out or give them to someone, than put in the time and effort to end up with a complete loss. Good luck.