Accessing Cards in Netrunner
Access in Netrunner is highly dependent on server location. The following draws primarily from the latest FAQ, and is supplemented by my notes/clarifications.
From the FAQ, with additional emphasis:
When accessing cards in Archives, the Runner turns all cards
faceup in Archives before accessing them. Then he accesses and
resolves individual cards one by one, in any order he wants.
So, the runner has a choice in terms of order of access (if there are multiple agenda points and 3 Shocks, they can access the agendas first and theoretically win the game before the Shocks flatline them), but they must access every card in the archive.
One note about accessing archives, if the Runner uses a replacement effect based upon a successful Archives run (such as Retrieval Run), then none of the cards there get flipped (e.g., the runner does not get a free peek at what is hiding there).
Accessing Multiple Cards from HQ
When you are given multi-card access from HQ, you must access your required number of cards (e.g., if you are successful making a run using Legwork, you must access 3 cards). You do so one at a time, and fully resolve each access before moving on to the next. Cards accessed in this manner do not return to HQ until you are finished accessing all 3 (so you pick one card from HQ, look at it, put it aside, pull another, etc.).
Accessing Multiple Cards from R&D
Similarly, when accessing multiple cards from R&D there are a few notes in the FAQ:
The Runner can intersperse accessing cards from R&D with any upgrades installed in the root of R&D.
The Runner cannot choose to access the cards in R&D out of order.
So, with multi-access, you always start with the top card of R&D and work your way down, fully resolving each before moving onto the next one (e.g., you can't fail to trash an installable asset, see the next card, and then decide to trash the previously seen asset). At any point during these accesses you can alternate between R&D proper and its root (in the case of upgrades installed in the server).
Accessing Cards in a Remote Server
When you choose to access cards in a remote server, it works in a similar fashion. Once committed to accessing, the runner must access all cards in the remote server, however they are able to choose the order of access (and it should be apparent which card in the server has been advanced, as well as which card was placed in the server most recently, if applicable). For example, the Corporation is not allowed to shuffle the cards in the remote server.
Circumventions of the Above Rules
While in the majority of cases the number of cards available for access is completely dictated by access location and board state (e.g., access all cards in Archives or in a remote server, access 1 card from HQ/R&D if running without any special card effects), there are a few cards that do give the runner slightly more leeway.
For example, Medium states: "Each virus counter after the first on Medium allows you to access 1 additional card from R&D whenever you access cards from R&D". This is not a required additional access, like the other cards described above. According to the FAQ, how this plays out is as follows:
Before accessing cards from R&D at step 4.5 of a run, the Runner
chooses how many cards he wants to access when using Medium.
So, after the run is successful (by passing all the ICE and deciding that you will access cards), the player must commit to the number of cards that they will access. If Medium has 7 counters on it, the Runner could choose any number between 1 and 7 - however, once that number is chosen, they must access that number of cards. There is no window to jack out after the number of cards has been declared.
This is in contrast to cards like Legwork, HQ Interface, R&D Interface, and The Maker's Eye, which all are mandatory effects (if run is successful, "access 2 additional cards" - there is no room to declare a different value).
Finally, it should also be noted that even if you are using one of these cards that have a mandatory effect for additional access, you do have a window at step 4.2 of a run to jack out after passing all the ICE but before committing to access. This can be especially useful if, for example, you face-checked a Komainu, and fear that some of the cards you access will cause you to flatline. For the purposes of other card effects, such a run would be considered unsuccessful.