If the source is an open-access paper under a creative commons license, it's easy - copy the figure and attribute the source and license.
If not... most publishers have a system for this. The STM Association, which covers the majority of mainstream scientific publishers, has a set of guidelines for permission between its members, setting out what is reasonable to reprint, and importantly has a general reciprocal agreement that they will honour each others requests without charge. This covers both books and journals, broadly speaking.
The STM list has the appropriate links per publisher. Some have a blanket permission, others require a bit of paperwork. For example -
- a figure copied from a Portland Press-published paper does not require permission to be reused;
- a figure copied from a Royal Society of Chemistry published paper does require permission, but this permission will be automatically granted at no cost if you fill in the online form correctly.
You do not need to do this before submission (at least, not in the cases I've dealt with) - it comes after acceptance when you know where it will be published and you're sorting out the details.
In the majority of cases, because copyright is transferred to the publisher (or they are given a very broad license to authorise republication) you do not need to approach the author. However, you might want to anyway.
Firstly, it's polite, and I'm sure they'd be pleased to know you're publishing something building on their work. :-)
Secondly, and potentially more importantly, permission to republish doesn't mean that the publisher will send you the files. You might need to get hold of a copy of the master images rather than the version put out by the first journal, in which case you'll have to speak nicely to the original author and hope they can find them...