Bikkurim, Chapter 1, Mishnah 3. This mishnah details which fruits are not subject to the laws of bikkurim, and from what time onward they may be brought.
"Ein mevi'in bikkurim chutz mishivas haminim" - bikkurim are brought only from the seven species:
A fruit that is not one of the seven species with which the Torah praised Eretz Yisrael - an apple, for example - may not be brought as bikkurim at all. Bikkurim are brought only from the seven species:
The two grains: wheat and barley.
The five fruits: grapes, figs, pomegranates, olives and dates.
Fruits that are not of the finest quality:
Even within the seven species, one brings only from the finest of them. The mishnah lists the following:
"Velo mitemarim shebeharim" - although dates are brought, one does not bring dates that grow in mountainous areas, since dates grow at their best specifically in the valley.
"Velo mipeiros she'ba'amakim" - the other fruits, such as pomegranates, are not brought when they grew in a valley, since fruits generally grow at their best specifically in mountainous terrain, dates being the exception.
"Velo mizeis hashemen she'einam min hamuvchar" - even from olives designated for producing oil, one does not bring those that are not of the finest variety.
The time for bringing bikkurim:
"Ein mevi'in bikkurim kodem la'atzeres" - bikkurim are not brought before Shavuos. "Anshei Har Tzevo'im hevi'u bikkureihem kodem la'atzeres, velo kiblu mehem" - there was an incident with the people of Har Tzevo'im, who brought their bikkurim to the Beis HaMikdash before Shavuos, and their bikkurim were not accepted from them.
The source for this halachah is the verse: "Vechag hakatzir bikkurei ma'asecha asher tizra basadeh" - the festival associated with the harvest is Shavuos, "bikkurei ma'asecha" means the first fruits of your labors, "asher tizra basadeh", which you sow in the field. From here we learn that Shavuos is the earliest time for bringing bikkurim.
True, the bikkurim referred to in connection with Shavuos are the two loaves brought as a meal offering - the first grain brought in the Beis HaMikdash. Nevertheless, since the verse calls them "bikkurim", we learn from this about bikkurim in general, including the fruits of bikkurim, that they are not brought before Shavuos.
And why were they not accepted from the people of Har Tzevo'im? By strict law, when someone brings bikkurim before Shavuos, they are accepted from him and the fruits are set aside until Shavuos arrives, and then they are brought in the Beis HaMikdash. However, these were prominent people, and there was a concern that others would follow their example. This is how the Rishonim explain it based on the Yerushalmi.
In summary: in this mishnah we learned that bikkurim are brought only from the seven species, and even from them only from the finest, and not from dates grown in the mountains, not from fruits grown in the valleys, and not from oil olives that are not of the finest quality. We also learned that bikkurim are not brought before Shavuos, as derived from the verse "Vechag hakatzir bikkurei ma'asecha asher tizra basadeh", and we examined the incident of the people of Har Tzevo'im, from whom the bikkurim were not accepted because of their prominence, lest others follow their example.