I would say that looseleaf green tea would be a big step up from teabags. A strainer basket that goes in a mug is a very low-fuss way to make looseleaf tea, as they are easy to clean and reusable. Looseleaf Japanese sencha was a game-changer for what I thought green tea could be.
I am personally not a huge fan of the mesh balls as tea tends to escape out from the gap along the middle, and for green tea especially, too much particulate can be bitter.
I also have a teapot with a strainer which functions much the same way, but for when I want to brew more than one cup.
I would get a good way to steep tea. Like a tea strainer or a dedicated teapot with a strainer. Then you can try different loose tea mixes top find what you like.
Adding on to this, I really like Adagio's IngenuiTEA 2 (I actually have the Teavana one, but they're the same thing). You just put your loose leaf tea in it, pour in hot water, and let it steep. When it's done you set it on top of your mug and it filters the tea out the bottom
Do not brew your tea in what you make you cofee as both beverage have a strong taste that affect the pot they are are sitting in with time.
If you brew coffee in a coffee maker, any pot with a top will do. You heat the water to the temperature indicated on the package, turn off the heat, add tea leaves and wait the time given. The leaves will have plenty of room to float and by not making tea in your mug, you make ot much easier to strain the tea leaves out: you pour the tea in a mug on which sit the strainer
You could also buy some tin box to store your leaves - especially if you only fond them in big quantities, and a tea pot to mix flavor as little as possible.
Find a Chinese or Japanese tea shop. If you live in a city, there probably is one. Get yourself a nice tea set--just a pot and some cups. You'll find a ginormous selection of teas there. Just try things out and see what you like.
You don't need gadgets! My on-the-go setup is a couple of disposable coffee cups and a strainer and with a source of hot water I'm all sorted. Brew in one cup, strain into the other, drink, repeat until the tea's all brewed out.