I used to fret about being able to read names, but, having lived in Japan for a while now, I’ve realized it’s a problem for everyone and they just deal. Especially as a foreigner, no one cares if you misread a name for the first, second, or even third time. In fact, reading it correctly is amazing! The best way to go is just try to learn the 訓読み of kanji even if WK or another source doesn’t bother (I learned the とう reading of 藤 long before I learned the 音読み).
As for my methods, living in Japan is the most obvious and easiest. I meet people all the time and read about more people in the news. I watch a lot of anime, and I make sure to learn the kanji for every character’s name. That alone, considering how strange names can be both in anime and reality, helps considerable.
I think the thing that bothers me most about Japan is the number of names that can be spelled the same but pronounced differently (花名, for example, can be はな or かな, and there’s really know way to know other than to ask or guess ).
Places I’ve never found too difficult to read, but trying to guess the kanji for names I hear about is always annoying.