Can you take in the mail on Shabbos?
Certainly some mail deals with matters which are inappropriate reading material for Shabbos and one would not be willing to use them for other purposes, accordingly this mail is muktzeh and cannot be handled on Shabbos. (See Mishneh Berurah 307:56, It is not clear to me that one is prohibited to handle mail which one does not know to be in such a category out of fear it is.)
Regarding other types of mail:
The Bi'ur Halachah (340:14, הניר ), as I understand it, explains that when a letter is enclosed in an envelope then according to everyone ripping it open is מקלקל (destroying) and only prohibited by Rabbinic law (though he continues to cite an opinion of Rashi which may render this not to be the case). HaRav Shlomo Zalman Aurbach zt'l challenges this opinion since typically destroying a "utensil" for the sake of extracting its contents is permitted (provided, of course, no melachah is involved). Insofar as/provided that one is no making a neat slit to use the envelope to store the letter but simply destroying it and discarding it, it should be permitted. (See שמירת שבת בהלכתה כח הערה טו). Rav Moshe Feinstein zt'l, on the other hand, agrees that opening an envelop is prohibited and therefore it's muktzeh (אגרות משה או"ח ח"ה סי' כא ס"ק ה, interestingly he notes with the advent of telephones the leniencies associated with reading letters which may contain urgent information is no longer relevant). [Of course this is the type of thing one really needs to discuss with their Rav]
Not all mail is enclosed in an envelope. If my subscription to Jewish Observer showed up a couple days late it would not require any tearing for me to read the articles. Such mail certainly does not become muktzeh (simply) because it is forbidden to open the (non-existent) envelope.
Regarding mail which is brought from outside the techum (the perimeter 2000 amos around the settlement one is located in, past which it is prohibited to travel on Shabbos):
An item which is brought on Shabbos from outside of the techum for the sake of a Jew is prohibited for that Jew to benefit from (until enough time has elapsed after Shabbos for the action to have been completed then). Other Jews, however may benefit from the item, and furthermore even to the Jew for whom it was brought the item is NOT muktzeh. (See O.C. 325:8 and Mishneh Berurah 307:56, though the above mentioned אגרות משה appears strict in this regard).
Certainly if it can be verified by local Rabbonim that the mail delivered on Shabbos arrives within the techum prior to Shabbos (or presumably when the mail originates within the techum and delivery is such that it would not be removed prior to delivery) the outside the techum issue is not applicable.
It would seem that there may be basis to bring in the mail from the mail-box (provided there is an eruv). If however there is mail which IS muktzeh mixed in with mail permitted to be handled it probably constitutes a teruvos (mixture) to which the laws of borrer (selecting) apply and one could only select the permitted mail, by hand, immediately prior to when one is going to read it. On the other hand, there might be basis to handle the entire mixture at once, especially if the majority is permitted (probably not though).
So it seems to me. There are a lot of different issues at play, many of which have a number of divergent views among the Poskim, certainly something to get a psak from your Rav about before acting l'maaseh.
Since you may not open mail on Shabbat, any mail that is sealed may not be taken in. If the mail came on Shabbat from outside the Tehum Shabbat (the 2000 cubit boundary outside of settled area) then all the mail is forbidden to be moved.
Well bills are muktza, other mail is not. Either way one cannot open a letter on shabbat if the envelope is sealed.
I've also heard some people say they don't bring in mail or newspaper on shabbat because of Nolad.