There is plenty of fairly good combat description in some of the long epic fantasy series... Sword of Truth, Song of Ice and Fire (Game of Thrones), Wheel of Time, etc. Sword of Truth features lots of magical combat, between small groups as well as large armies, combat description often includes lots of gore and unpleasant scenery. Song of Ice and Fire tends towards more realistic combat, with magic substantially less prevalent, actual combat description is somewhat less frequent than some other series. Wheel of Time features much more sanitized combat with less gruesome scenery. Personally, I particularly enjoyed a lot of the combat description in Sword of Truth.
Another series which has some good magical combat description is the Name of the Wind; it's combat scenes are limited but they are of a high quality, in my opinion. Name of the Wind has a magic system whose function is described in great detail, almost as a science.
The Lord of the Rings, which you mention, is a great novel primarily for the depth and intricacy of its backstory and the world within which it is set (as described in the Silmarillion and other books), but it is not particularly "action-packed" and has only a few scenes of combat throughout, focusing much more on the description of the world of Middle-Earth as the characters travel throughout it.
As an aspiring author, it would probably do you well to read at least a few of the major works in the genre you are attempting to enter...
If I had one general tip about writing combat scenes that are engaging, I would say that the main thing you have to do is to make it seem real and make it touch the emotions of the reader, usually done by evoking the emotions and sensory perceptions (all 5 senses or more if applicable) of the main characters, that the reader (hopefully) has an emotional investment in (either identifies with them or hates them). Having an emotional impact can be done by illustrating tragic events/situations, through visceral scenery, by demonstrating heroism/bravery in the face of hopeless situations, etc. Stay away from dispassionately cataloging a mere list of combat events, unless you are writing a history book.