In Kol Mekadesh that is sung Friday night it says המאחרים לצאת מן השבת וממהרים לבא which seems out of order. Do we not first rush into Shabbos and then delay its departure? Then why do we sing it in the opposite order?
I once heard an answer as follows: The Torah was given on Shabbos, so the first extension of Shabbos was the מאחרים לצאת they did that evening. It was only the following week that they fulfilled ממהרים לבא.
A similar answer is that at the time the song is being sung, it is already Shabbos, so the next event that will occur is מאחרים לצאת, not ממהרים לבא.
Another approach is that one may have ulterior motives for beginning Shabbos early (e.g., he is tired, hungry, had a long week, etc.). But to extend the ending of Shabbos truly shows one's respect of the day. Thus, we first say מאחרים לצאת to prove that yidden truly value Shabbos, and then the ממהרים לבא is understood to be for Shabbos' sake, rather than for one's own benefit.
[However, let me also point out the obvious: לבא rhymes with אוהבו and the other words at the end of those stanzas; לצאת would mess up the poetry. Just saying!]
I once heard an answer that throughout the years in the desert, the Jews always knew exactly when Shabbos started and ended becuase they had the pillars of cloud and of fire. However, these were taken away when Moshe died, which was on Shabbos. That night, they did not know exactly when Shabbos ended, so they delayed ending it until they were sure. The next week, they started early for the same reason.
Beit Aharon: When Bnei Yisrael got the Shabat at Marah, they didn't know its taste. Then when they first tasted it, it was so good that they didn't want to leave it. Then, they waited all week for the next Shabat and they hurried up to get into it.
The week begins when Shabbos ends. It would seem to me that this is just giving the chronological order of events.
I heard this explanation from Harav Mordechai Kamenetzki shlit'a:
This is to convey the idea that Shabbat occupies ones entire being throughout the week. As Shabbat is about to end, people should be so devoted to Shabbat that they hate to relinquish it. So they delay its leaving.
This is stated first to keep, in a sense, with the concept of סוף מעשה במחשבה תחילה which could be translated as "The end, which is in action, begins with the thought of it at the beginning." I.e., the action of delaying the ending of Shabbat is thought of at the beginning of Shabbat.