First up I fully second the recommendations for
Robin Hobb's work - I'd suggest starting with Assassins's Apprentice. Whilst each of her book series stands alone they run one into the next and if you skip on you are likely to miss out on some key bits of info or characters. Rather more mature writing here overall.
Temeraire - dragons and cannon and warships. Lot of fun in this series and well worth reading if you're a dragon fan
The Hobbit and the Lord of the Rings - The Hobbit is far more accessible and easier to read so is a great starting point; LotR is a bit heavier in content but equally a great story and one of the big inspirational works of modern times. There are also several editions of both with great artwork in them if you hunt them down.
Seraphina - As said earlier this is a very unique and different take on dragons. I enjoyed reading it and haven't yet delved into the second book
His Dark Materials - Great fantasy adventure and far far superior to the film that was made.
Sabriel - a really powerful opening book and a great start to the series (I felt the first two books were great, the third a bit weak and eager to finish things a bit too fast).
And now for new suggestions, I'm no good at telling where young adult starts and finishes though I'm sure @
kathyy would be able to give a better guideline on that score if one is interested
1) Dragonflight by Anne McCaffrey - aka the "dragon lady". One of the longest series of books focused on dragons there is and a major work in its day and still today. You might find the writing a little more of a challenge to read than more modern books, but the style is easy to get used too once you get into the story.
2) Dragon Champion by E E Knight - one of the fewer books out there where the lead character is a dragon. No dragon riders, dragon tamers etc...; in this first book of the series you follow a single dragon. The following books start off following others from the same clutch of eggs and how these dragons interact and shape the world around them.
3) The Colour of Magic by Sir Terry Pratchett - the beginning of the huge Discworld series which focuses on a mad world of magic wizards witches and more. Witty and intelligently written and great fun. Each book stands on its own, but if read in order they carry links; you'll see familiar characters, notice running jokes and themes and also see the general huge strides the Discworld takes in, well, modernizing!
4) Legend by David Gemmell - mighty heroes, impossible odds and a fortress at siege. High sword swinging adventure with a touch of magic. Gemmell is a great read and this is the first of a series of similar books.
5) Daggerspell by Katharine Kerr - the start of a long running series of books. Set in a mythical realm that gets ever more magical as the series progresses. With a heavy influence of celtic background to it. Welcome to a world of round castles, roving warriors, dispossessed princes and more. A bit of a complex series in how the characters work, but a great one to be sure
If you've got an Amazon account (and especially if you use an e-reader, but you can read them on your computer as well) then I'd recommend a subscription to "Fantasy and Science Fiction" magazine. It's very cheap and you get a choice selection of short stories every few months. A great way to get some varied short fantasy and sci-fi stories; many a big author has often started with such short stories so you can find new authors to look up.
If you're after more mature reading in fantasy then I'd suggest
1) A Game of Thrones by George RR Martin - its only downside is that its not finished (yet) but otherwise is an epic world of low magic but epic fantasy with a huge roster of mortal characters.
2) Gardens of the Moon by Steven Erikson - like game of thrones this has huge armies, many many characters and epic events; however this series is both finished and also complete with a huge amount of magic, gods, spells, inhuman characters and more.