What sort of saga are you planning/playing and what sort of magic and adventures interest you and your players? The list of books changes a lot depending on that.
Like Xavi said, it depends heavily on what your troupe is into.
My list would be as follows:
First: HoH: S and HoH: TL
Second: the Tribunal book for the Tribunal your Saga is set in.
Third: sit down with your players and discuss their interest in the four Realms (Magic, Faerie, Infernal, Divine). Ask them which Realms they would like their stories to touch on. That should give you the priority in which the Realms books should be bought. If you've got a player who threatens to quit the game if you involve faeries, you can probably put off buying RoP: F for a bit. If your entire group consists of atheists and dyed-in-the-wool pagans, you're probably not going to need RoP:D for a good long while. If you've got a bunch of fledgling demon-slayers on your hands, you need RoP:I post-haste. You get the idea.
After that, you can play it by ear, depending on the age of your Saga and the power-level of the magi. (Don't run beginning characters through BCoC, for instance: I did that recently, and had to fudge like mad to avoid the entire party getting slaughtered. )
For me, Mysteries and Ancient Magic were a must have.
I own Cradle and the Crescent as well as Rival Magic, and neither have proven to be useful in a saga just beginning. They are a must for well established stories and characters though, IMO
I think Arts & Academy is a excellent book, as is Lord of Men and City & Guild, but I think you can do without them all if you are running a looser game.
I can happily run a game in Normandy and the Rhine without ever once cracking their Tribunal books.... but the Theban and Transylvanian books are a must have if you run a story there. ((and once you read either of those books, I place a bet you will want to run a story there ))
In the end, for me most of the Realms books can be ignored as they lead to some cumbersome character generation rules, but that is because I am feeble minded. Maybe the realms of power magic book....
Here's what I feel those books offer you, beyond the obvious:
HoH:S - Imaginem & deception, pairs well with C&G craftsmen or A&A artists
HoH:TL - Breakthrough & Tribunal Law, the least useful House book imo
RoP:M - creature creation mecanics
RoP:D - non-christian Abilities, pairs well with The Church
RoP:I - munchkin tool
RoP:F - story theory
MoH - good spell/item source
All the House books. The tribunal book that you will run in. I would slowly pick up other Tribunal books because they are good background. Mysteries Revised is also a really good. Two recent books that have been very useful have been Mage of Hermes and Hermatic Projects. I feel the realm books have useful information but to read through once. RoP:M has been the most useful for the Beasts of virtue for familiars. The others were read once and put on the shelf.
I am Tertiary SG / ideas monkey for one group (I don't play with them anymore, I just gave them a how-to assist to start them off.). That's where some of the rules clarification queries come from.
I am Alpha (and only) SG for another group where I am using ArM mechanics, but a seperate universe (not M.E.).
I actually get to play with a bunch of code-shattering experts, who are aided and abetted by darkwing (Beta SG) and cj.23 (Alpha SG). <It's quite worrying when you are telling other roleplayers and SGs about your group and the shocked expression just gets worse as you try to explain it.>
I ended up getting LoM purely for the crossbow rules. Yes, I am that bad.
I have since instigated a rule that if the player wants to play it that badly, they can buy the book for the group. Working suprisingly well. Thanks to that little brainwave, Our group collection of Ars is quite reasonable. This list is purely for me, so I don't have to run back and forth to check on things as I work.
I was thinking of completing the House books first anyway, so that makes sense.
Tribunal books... to be honest, I'd like to see an actual stock tribunal first, all the ones I've seen have been variations on the norm. Besides, my SG style is to use as few books as possible anyway. (actually my SG style in one game I'm running is to turn up without even pencil, rulebook, dice etc. and just run the story.)
Tried that... I believe it went something like "Magic, Magic, Magic, Divine." Entertainingly, as a "dyed-in-the-wool pagan" I am actually quite fond of Holy Magi.
I actually found it better to ask them what kind of characters they wanted to play, the "Divine" answer was because she wanted to play a Paladin-type, and hunt demons.
To be honest, TMRE and AM are in the list, mostly beacuse I keep needing to refer to them to fix a problem in another game.
Tugdual, thankyou that was actually very helpful. A&A and RoP:M are definitely in my next 5 anyway, it's just the next 1 that's the problem!
Would you put MoH, Grogs, or HP first on your list? I can afford to buy maybe on book a month at the moment.
That's something I've noticed TBH. RoP:M gets used a lot, and the mechanics are fairly easy to get your head around compared to the others.
For a standard tribunal get Heirs to Merlin (stonehege tribunal). It is 4th edition, but has no stats at all, so it is still 100% valid for 5th edition. It came out just before 5th edityion and it is authored by David Chart himself, so you can see a pattern for "standard" there Next tribunal I would be expecting will be Provençal or Hibernia (the 2 only uncovered tribunals so far). I doubt that Hibernia will be standard (it is in the fringe) and Provençal is likely to be heavy in chanson de geste romance and albigesian crusade stuff, so not traditional either. This is what I dislike about the Normandy, that IMO should be as standard as you get and instead it is totally non standard, but that is me (rant: off)
House books are useful.
ROP: Infernal is amazing for antagonists. The mechanics are IMO much more solid than with other books, and they integrate in the common mechanics much better than those in other books. Plenty of stuff on bad guys. Not all of them blatantly infernal, and this is a good thing. The Ordo Vagorum played quite a role in our saga. We almost had a heresy (and crusade!) right in our doorstep. The other realm books have been much less prominent in our saga, even if we use fae and stuff like that all the time.
I have had GREAT fun with the 3 hedgies books (HMRE, Rival Magic and Ancient Magic, that is a hedgies book as well as a story hook book). I would only recommend them if you are into non hermetic stuff either for your (human) antagonists or you want to use them as PC.
Tales of Middle Earth and Magi of Hermes are great for NPC and location descriptions right on your lap. Magi of Hermes has been great to populate tribunals, ally and enemy places, sensibly reducing our alpha SGs (read: me) load of work. ToME was just the core for the development of our whole last saga (along with the excellent Hermes Portal tribunal book on the isle of Mann). Of these two and Legends of Hermes I would put first Magi of Hermes. A review on Legends by yours truly can be found here: amazon.com/review/R271OXBWW5YAY2
Whenever I'm looking for ideas for Magi or how to do something I always pick up MoH. That is from a player perspective. From a SG perspective Grogs is better if the campaign is for magi 1-5 years out of gauntlet. Hermatic Projects is interesting for big ideas.
As has been mentioned before, the complete "Houses of Hermes" series is worth getting as it adds a lot of flavor to the individual houses and also has lots of mechanical benefits for players to toy around with.
I'd also strongly suggest getting the complete "Realms of Power" series. In my saga I found I've been using Infernal and Faerie by far the most, followed by Magic, and Divine seing rather rare use - while it contains some essential information, it's not a book that is often referenced.
For me these two series are the "core" of the ArM5 supplements. After that it's mostly a "depends on what you want" scenario.
Mysteries is an interesting book and a rather nice tool to introduce secret societies and give players an option to gain Virtues in the course of the game (other than Warping). Most of the actual Virtues presented therein tend toward "meh" though (IMHO).
While I do own "Art & Academe" and "City & Guild" they never see any use in my saga, since I feel they complicate their subject matter too much - this is probably fine for a saga focusing on their themes, but mine obviously does not.
The one book that I wasn't actually planning on getting at first, but wound up using the hell out of is "Covenants" - lots of useful and interesting stuff that I mostly didn't expect to be interested in.
And as has been mentioned: Get a tribunal book to get a feeling for an established setting. You don't have to stick to it, but you get the feeling for how things might come together. I'd suggest "Guardians of the Forests", since it's a rather traditional tribunal and therefore can serve as a good foundation for modifications.
"Ancient Magic" is very specific and only relevant if your players are planning on expanding the limits of Hermetic Magic. And even then you probably don't want to use more than one of the options presented in this book per saga.
Finally, "Hedge Magic" and "Rival Magic" are good sources for antagonists (or potential allies, depending on your bend), as is "Cradle & Crescent" to some degree.
))
Thanks to that little brainwave, Our group collection of Ars is quite reasonable. This list is purely for me, so I don't have to run back and forth to check on things as I work.
Next tribunal I would be expecting will be Provençal or Hibernia (the 2 only uncovered tribunals so far). I doubt that Hibernia will be standard (it is in the fringe) and Provençal is likely to be heavy in chanson de geste romance and albigesian crusade stuff, so not traditional either. This is what I dislike about the Normandy, that IMO should be as standard as you get and instead it is totally non standard, but that is me 
(rant: off)