There are lot of recommendations on what endgame books you should read and study. But reading is just passive learning, and not so efficient.
If the book doesn't contain exercises, what is the best method for active learning and doing exercises?
One idea I have is to play the positions against an engine. Are there better suggestions?
Personally, I love a number of great books, and there is still no substitute. You can then plug positions into a computer to test yourself.
I still love Fine's "Basic Chess Endings". Despite the errors, many corrected later in the Pal Benko edition, it gives a great sense of WHAT you are trying to accomplish in any type endgame. Simply by reading over all the examples, and the sub-examples, you will absorb a lot. Each sub-example, while meant to show an exception, or other deviation, is also reinforcement of the main idea.
For example as to my comment about "WHAT" you are trying to accomplish, take a minor piece endgame where you are up a pawn. The basic plan is to create a passer, and if your opponent blocks with the K, you run with your king to attack and take his other pawns. It he just tries to block it with the piece, you run over and force him to give up the piece for the pawn.
I studied this book when I was rated just 1036, and much of what I learned has stuck with me through being a Master almost 40 years later. No less than World Champion, Mikhail Botvinnik, considered it the best book on the endgame (although many greats ones have come along since). Fine's words are chess gold.
Next, I really like "100 Endgames You Must Know: Vital Lessons for Every Chess Player" by GM Jesus de la Villa. Not many books get a fourth edition, which is a testament to this book's appeal. In addition, he recently added "The 100 Endgames You Must Know Workbook: Practical Endgame Exercises for Every Chess Player", which lets you apply the 100 endgames from the first book in practical game exercises.
Lastly, if you really get strong, I still consider "Dvoretsky's Endgame Manual" THE single most instructive book ever written. That said, it is not for novices, and expects a lot of prior endgame knowledge. Similar to de la Villa's recent workbook, Dvoretsky also wrote "Tragicomedy in the Endgame: Instructive Mistakes of the Masters", which shows practical games involving information from DEM.
No matter what you do though, endgame study requires time and effort; however, unlike opening study, this study will stick with you for a lifetime.
I've practiced some endgames against the computer, and I think it has been useful but there are some caveats. It may depend on the engine (and the endgame), but I've found that sometimes when practicing a won position or defending a drawn position, the engine does not play "realistically", making the endgame unnecessarily easy to win or draw.
Let's use KP vs K endgames as a simple example. Let's say you have a won position. A stubborn human defender would try to block the pawn til the end, hoping maybe you'll slip and allow a draw. The engine may decide instead "I'm lost, but if I run away with my king now, it will be mate in 12 instead of mate in 10!", so the king runs away and makes the promotion trivial.
Similarly, if you are defending a drawn position, a stubborn human would try to keep the pawn and advance it until stalemate is reached. The engine concludes there's no difference, so I've seen it give away the pawn, leading to an immediate draw.
I've also practiced endgames with a coach, and it's much better, because the coach knows how to make tricky moves which give you the opportunity to make a mistake, instead of just trying to maximize the distance to mate.
Actually in a case of endgames you can split learning in two parts:
You have to start with theory - read books - those will explain ideas and principles. Then You can move on (can do it somewhat in parallel) to practice. There are lot of internet sites, PC and mobile apps for that - I will not advertise any specific - that would be unfair. You can even just set-up position and play Vs your engine until you succeed getting desired result(win/draw). Also - learning is never easy - if you want to achieve high level You will have to work hard, spending many hours on this!
Are there better suggestions?
I find videos to be more effective than books and ideally the videos, as well as giving instruction, should also have problems related to the matter as you suggest.
My personal preference, which I find very effective, is the two DVDs on endgames produced by the Ginger GM, Simon Williams. The material is presented by English GM Nick Pert and covers everything you are likely to need from complete beginner (K+Q v K) through to GM level (K+R v K+B, K+h+f pawns v K, K+R+a,f,g,h v K+R+f,g,h, K+Q v K+R). You can find more details here.
I don't know about 'better', but solving chess endgame studies can be an alternative way.
Best introduction to that field is John Roycroft's book The Chess Endgame Study.
As for use of computer ... I'm in two minds. Yes, it makes it possible to get a slightly higher degree of realism, including the possibility of blunders (if your program allows you to set up that), but I think you also have to learn to see the position from the opponent's view ... and use of computer may not help very much with that.
