Learn Spanish (Latin America): El doctor mira las manos.

Usually, yes! Words ending in -o are mostly masculine. However, la mano (the hand) is one of the most common rule-breakers in Spanish. Because it is a feminine noun, we use the feminine plural article las, making it las manos. You just have to memorize this one!

In English, we say "looks at the hands". Why isn't there a word for "at" in the Spanish sentence?

In Spanish, the verb mirar means "to look at" or "to watch". The "at" concept is already built right into the verb, so you don't need to add an extra word for it. You just say mira las manos.

Does las manos mean his own hands, or just hands in general?

It could be either! In Spanish, when talking about body parts, you almost always use "the" (el, la, los, las) instead of possessive words like "his" (sus) or "my" (mis). Based on the context, he is likely looking at a patient's hands, or maybe his own.

How would this sentence change if the doctor was a woman?

You would change the masculine el doctor to the feminine la doctora. The rest of the sentence would stay exactly the same: La doctora mira las manos.