You have several options. The easiest is to buy a larger tea mug and use a large infuser basket that just fits inside it. Thats the first step up from teabag hell.
If you haven't had loose leaf tea before, you're in for a treat! Please considering clicking on the Adagio logo in the upper right-hand corner of your browser window to take you to the storefront site, so that can begin familiarizing yourself with the many excellent teas available from our host.
The next step up from a mug is to use a teapot that is designed to brew looseleaf tea. The simplest design is a plastic pot in which you add tea and water, let it brew and dispense it through a port at the bottom directly into your cup. That's the IngenuiTEA, mentioned by EB, and made and sold by our host Adagio. About as convenient as an infuser and cup, but a step up because it makes full use of the vessel volume for infusing larger leaved teas. You can purchase the ingenuiTea as a starter kit.http://www.adagio.com/gifts/holiday_ingenuiTEA.html
Your next option takes a bit more effort. It's a small 2-cup inexpensive porcelain teapot. You can purchase your pot with ceramic or stainless steel infuser baskets, you can purchase a plastic-and-nylon mesh version if your new teapot lacks an infuser, or you can go the old-fashioned route and use a small, handled strainer to remove tea leaves at the spout when you pour up your tea.
A little extra effort can yield a wonderful cup of tea, typically far better fare than can be had with inexpensive teabags. The cost isn't much higher, nor the effort much more than using a teabag.
Don't worry about unfamiliar teaware terms, like 'gongfu' or 'kyusu'. Let's keep it simple to avoid confusion and start with small steps.
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One more thing: there is a downloadable free tea timer that sits on your computer desktop. On the Adagio website, it sits just under the teaytpe banner at the top right of the home page. Once you download the timer and open it, you can select from a wide assortment of classic tea blends and pure leaf teas, set the timer and hit 'start'. You want to avoid overbrewing your tea to keep it from tasting bitter.