What is needed to play or run the game?
As a player, the Player's Guide should suffice if you are familiar with the system that your group is using. At present, Achtung! Cthulhu has versions for Call of Cthulhu and Savage Worlds (these are presented together as a dual-system set), PDQ, and Fate. If you are not familiar with the system, having access to it will definitely help. As a player, you will not need the Keeper's Guide, and reading it may have a detrimental impact on your experience.
As a GM, you will need both books (Keeper's and Player's Guides) and the core rules for the system. There are now several scenario and campaign books which extend the setting to other theaters of war which may prove useful also.
The main version of the game as printed is compatible with two systems, Call of Cthulhu and Savage Worlds. Other versions exist to support different systems, such as Fate, singly.
Call of Cthulhu
The main version of the game states it is compatible with Call of Cthulhu 6th Edition. That actually means you can run it with any edition of the game from 3rd Edition onward with no conversion (1st and 2nd lack some rules coverage you will want for a military-focused game), but they are compatible also. Some conversion will be required if you wish to use the forthcoming 7th Edition.
This compatibility also means that with familiarity or a little conversion you can run the game with Basic Roleplaying (BRP), RuneQuest 6th Edition + the Firearms Guide, OpenQuest, or Renaissance Deluxe.
The game as broadly released also offers compatibility with Savage Worlds Deluxe Edition. The GM will need this book and again whichever of the additional setting books they deem important.
The other large-scale release covers running the game using Fate. The GM will need this book and the Fate Core rules, as well as whichever of the additional setting books they desire.
Should Call of Cthulhu be played first?
There is no simple answer that everyone will agree with, so in the end - much like a protagonist in a Lovecraft story - you will face the horror of decision on your own.
Call of Cthulhu: Many settings and approaches, one game... mostly
Call of Cthulhu is an enduring, classic. It pits characters against deep mysteries that threaten body, sanity, and soul. Run well, it is a gaming experience that can enhance the way that you game and perhaps even change the way that you see gaming. Run poorly, it can be just another monster hunt.
There are many different takes on Lovecraft & Friends' Cthulhu Mythos in roleplaying form - even within the Call of Cthulhu (CoC) brand. For example, the original game itself has been divided along era lines from very early in its history, featuring support for play in the 1920s (standard era play), Gaslight's 1890s era play, and Cthulhu Now! for the modern era. It has seen other expansions and revisions as well taking the game backward in time to the Dark Ages and to Roman Times (Cthulhu Invictus). During the age of the X-Files, some of Lovecraft's more aggressive stories were combined with the interest in conspiracy, shadow governments, and mysterious beings among us to produce a cold-war meets Cthulhu game called Delta Green. Each of these games retains an aspect of the original approach, yet offers different roleplaying opportunities and challenges.
Licenses of the Game
Licenses of the brand bring us directly to your question. Taking the game and putting it in a modified setting and/or with a different game system also have an influence on how it plays. The play experience can be changed by how the Investigators are represented mechanically. This affects how dangerous each scenario feels and changes the relationship of player decision-making to character risk. The experience can also be changed by the setting's take on the prevalence or activity-level of the cosmic threats in the game. Games like Achtung! Cthulhu (A!C) fall in this band. The game can be played in a variety of official system adaptations from the original CoC system (Chaosium's Basic Role-Playing; BRP) through PDQ and Savage Worlds. Each has its effect on the fragility of characters, and places different emphasis on that fragility. More important to your question, however, is the effect of the setting on your experience as a new player.
A!C intends to frame its characters' struggle against supernatural threats and human agents in the context of a military action genre. Investigation is a core part of the game, but so are big guns and battles with strange creatures from beyond. This is a true part of Lovecraft's writing (Check out what happened in the Shadow over Innsmouth for example), but not a primary part of it. As such, there are some CoC players who may feel that this 'pulp style' play is not being true to the heart of this roleplaying game's lineage. Such players may feel then, that for a 'pure' experience, you should start with CoC first, then expand from there.
In my opinion, there are a lot of things which can be done with the Cthulhu Mythos, the writings of Lovecraft, and that of those he inspired. I feel that CoC is a fantastic game, which had profound effects on my gaming both as a player and as a GM. Were I to introduce a new player to a Mythos game, I might very well choose CoC as the medium for that experience. That said, A!C is a well-written setting with a clear focus on its goals for play. I would love to play it with experienced CoC players who were looking to find a way to bring the horror of the situation to light in their roleplay, over the imagined roar of big guns.
Ultimately, as a player, what kind of experience in what sort of setting do you desire? Do you wish to explore the future of humanity among alien stars (Eldritch Skies), do you wish to investigate corruption in the drawing rooms of the elite (1890s) or in the dry streets of prohibition-era America (1920s)? Do you wish to explore the horrors of our modern world, with the knowledge that the vast evil we do to ourselves is pales in comparison to the blind malevolence which awaits us just outside the range of our paltry human senses? Do you enjoy military action in a grim but respectful WW2 setting, which poses that the desperation of war might make the enemy willing to truck with entities it has no hope of controlling, as the world around us is shaken to its core by the never-ending roll of steel and artillery? The answer to any and all of these might be 'no.' You may find that the idea of merging CoC with the first or second world war appeals to you, but the pulp elements which can be a part of A!C appeal to you less. In that case, you might consider World War Cthulhu instead. Or, for you, the answer might be to forgo all of these settings based on our world and instead explore the game via the Dreamlands, a curious place beyond the veil of sleep, but no less threatened by ineffable cosmic forces. Of course, with the ongoing popularity of Cthulhu, there are many more options than just these, in many different genres... even giant robots...in space.
There is much good gaming to be had, whatever you choose. That you would ask, suggests to me that it is that so-called, 'pure experience' that you desire first. I do not think you would be disappointed were that what you held out for... just do not hold out so long that you miss your chance to play at all.
Essential things for a new player of Lovecraft are simple. Read the short story, 'The Call of Cthulhu,'by H.P. Lovecraft and a few others, such as the aforementioned 'The Shadow over Innsmouth' and 'The Dunwhich Horror.'
This done, open your mind to the possibility of, and more importantly enjoyment of, experiencing the terrified viewpoint of a Lovecraft character. The world around you is not what you thought... it's much worse than that... it always has been... and it won't notice at all when it shreds your mind and shatters your dreams.
With this viewpoint in mind, it doesn't matter which game you choose, so much. Just play that choice well, and hopefully you will have the opportunity to do this with others who take it as more than an excuse to pretend to shoot things.