A friend in North Carolina recently showed me a yellowing nengajō (年賀状, New Year's card) I had sent her soon after first arriving in Japan back in the early 1980s. The return address, penciled in my best effort at the time — a childlike, uneven scrawl of kanji — reminded me of the intense determination I felt then to become proficient in Japanese.
Despite the wizardry of today's Japanese word-processing software, I regularly write my address by hand. Many foreign residents of Japan from a non-kanji background always use rōmaji (ローマ字, the Roman alphabet) to write their address. If writing your address in nihongo (日本語) is intimidating, here's some advice on getting started.
Before going public, practice at home on lined paper. Don't be overly concerned with aesthetics: Aim for legibility at first. Take advantage of every opportunity to write your address: on forms from your child's school and at your workplace; on envelopes and takkyūbin (宅急便, home-delivery service) labels; at doctors' offices, the bank, city hall or video-rental shops; when you apply for a driver's license, cell-phone service, a train pass or cable TV. If necessary, carry a cheat sheet in your wallet. If you find a multi-stroke kanji particularly difficult to produce, write that character in hiragana.