How do I read manga?

With a little bit of practice, you'll get used to it. After a few books, you'll be surprised that you even felt you had to ask the question.

But, since you asked, here's the answer.

Start in the upper right. The general way to get through the balloons is work down then left.

With the panels as well, it's generally down, then left. But don't "force" it. If the panel layout makes it seem natural to go left rather than down, go left. Some inexperienced manga artists make confusing layouts, but most of those are caught by the editor before they get to the finished product.

If you still become confused at times, don't get too frustrated. I've been translating manga for over 25 years, and I still come to panel or balloon layouts where I have to explain to the editor and letterer what balloon comes next (and there are times I'm not 100% certain myself)

The same top-to-bottom way you'd read a western comic, only moving from right-to-left instead of left-to-right. So, the first caption or word balloon will be closest to the top right of the furthest right panel and your reading order progresses down and left from there. At the end of each panel row you start the next from the furthest right panel progressing left.

Be aware though that it's only in the last decade or so that English translations of manga have preserved this right-to-left reading order - editions from the 80s and 90s used to be 'flipped', artwork and all, for left-to-right reading order. Some of these are still available, like the Dark Horse 'Akira' paperbacks, or even the contemporary Drawn & Quarterly editons of Yoshihiro Tatsumi's gekiga works ('Abandon the Old in Tokyo', etc.)

You can obviously tell which reading order the publisher's opted for by seeing where they've placed the front cover - if the spine's on the left (when the book is shut) the manga has been flipped to be read left-to-right, and if the spine's to the right it's printed in the proper reading order. But this can be confusing when reading a digital edition where's no spine to clue you in - especially in those manga that open with a string of silent panels.

Presumably this is why DC have resorted to numbering panels by order in the first few pages of all of their Jiro Kuwata 'Batmanga' releases.

Reading Manga Sites are: MangaPan and 1st Kiss Manga