Timmo, I have been reading Tolkien since around 1967. Here then is one seasoned fan's suggestion:
Read the Tolkien books in publication order.
That means that, now that you have read
The Hobbit and
The Lord of the Rings, you would go on to read the delicious book of poems,
The Adventures of Tom Bombadil, first published around 1960. If possible read it in an edition illustrated by Pauline Baynes, who worked with the author on the project.
Then get hold of the music book,
The Road Goes Ever On, and read Tolkien's appended material -- just a few pages about Elvish and the most ancient history of Middle-earth.
Go here to read a rare Tolkien treasure that many people have never seen, Tolkien's interview with Henry Resnik in the fanzine
Niekas:
eFanzines.com - Niekas
You can find two light-hearted poems, "The Dragon's Visit" and "Once Upon a Time" in Lin Carter's 1969 anthology
The Young Magicians. Whether they are Middle-earth items or not might be debated.
There's a poem,
"Bilbo's Last Song," which was released in Tolkien's lifetime as a poster. It is now available as a book.
Having read these things, you will have read nearly all of what we had to go on up to Tolkien's death in 1973.* These are the things that he released during his lifetime. If you like, you might also consult the indispensable volume
Letters of J. R. R. Tolkien, which contains many nuggets of Middle-earth lore and reflections. However, only a little of this material was intended by JRRT for public dissemination.
You can read two delightful short works any time:
Farmer Giles of Ham and
Smith of Wootton Major, which are not part of the Middle-earth mythos.
After this point, you may read a couple of works edited by Christopher Tolkien, JRRT's son. I myself see these as "canonical," while recognizing that Tolkien might well have changed his mind about some things had he lived to work more on them. The two works I have in mind are
The Children of Húrin and
The Silmarillion. I would recommend the former as the one to read first, since it is more accessible, and then the latter, wherein you will find the story in
Children told concisely.
After that:
Unfinished Tales and the 12 volumes of
The History of Middle-earth. Chronicles' own J. D. Worthington has, I believe, read the entire set, and I hope he will comment here. My suggestion is that you begin with a treasure:
Athrabeth Finrod ah Andreth (in the book
Morgoth's Ring). The HoME set also includes items such as some late ponderings, for his own benefit, by Tolkien, of topics such as the origin of Orcs, etc. Two unfinished novels:
The Lost Road and
The Notion Club Papers may be found in the volumes.
I hope you will consider my recommendations seriously. These are wonderful things, but I think the order I have suggested is probably the best way to approach them.
*I'm not mentioning fugitive bits that appeared in newspaper articles, magazine profiles, etc., likely as not with reporter errors, such as that Tolkien was working on "The Scimarrion" or whatever it was that one poor journalist mentioned....