Am I right in thinking the greater the mAh, the longer the time a battery's power will last before it needs recharging?

In theory, at the same voltage and discharge rate, yes.

'In theory' because, for a given size of the battery (i.e. battery type) and a given battery chemistry, the capacity ('the mAh') is more or less fixed, which doesn't prevent some (mostly Chinese) manufacturers/dealers from claiming otherwise....

'More or less fixed' because there can be some trade-offs between different characteristics of a specific battery type. Capacity is one of them, self-discharge (how long the battery holds a charge with no load), internal resistance (which defines the maximum discharge current) and lifetime (how often can it be recharged) are a few others. For example, Eneloop (and other LSD NiMH batteries) sacrifice some capacity against a low self discharge, which makes them more useful in everyday household use.

Aye.