As far as the actual practices of the Three Weeks and the Nine Days are concerned, we prefer not to rule definitively, as the practices depend very much on family and/or community minhag. These are best handled on the local level. However, it is worthwhile to address some of the confusion about the source, rationale, and parameters of this halacha/minhag.
The issue is actually not the eating of the new fruit, per se, but the fact that when eating new fruit, one is required to make the beracha of Shehecheyanu. The wording of that beracha implies that we are happy to have reached a particular period of the year. However, the sadness associated with the Three Weeks makes such a statement inappropriate then. The source is actually post-Talmudic (as is not uncommon regarding these halachot). The Sefer Chasidim[1] says:
There were pious ones … who would not eat any new fruit between 17 Tammuz and 9 Av, for they said: “How can we make the blessing that ‘He gave us life, sustained us, and allowed us to reach this time’?” There are those who make the beracha on new fruit when they came across them on the Shabbatot between 17 Tammuz and 9 Av.
The Shulchan Aruch[2] phrases it as follows: “It is good to be careful not to say Shehecheyanu on fruit and clothes during the Three Weeks, but on a pidyon haben you say it and do not lose out on the mitzva.”
The weak language of these sources implies that the practice of avoiding the recitation of Shehecheyanu is in a lower category than halacha or even a standard minhag. The Gra[3] calls this practice an extreme stringency. He proves that even a mourner on the day of a parent’s death may recite Shehecheyanu if the need arises, and, therefore, there is no way that the Three Weeks could be more severe. However, the Magen Avraham[4] makes a fundamental distinction between the Three Weeks and mourning, which many accept (but the Gra does not). He says that it is not that a person should be too sad to recite Shehecheyanu but that the period is a tragic one. That is, the Three Weeks is an objectively sad time for all, as opposed to the occasion of mourning, where the individual is sad, not the period in time.
The aforementioned sources do not say that to make a Shehecheyanu during this time is out of the question, but that the situation should be avoided by not eating new fruit or wearing new clothes. (For clothes, there are additional problems during the Nine Days, beyond the issue of Shehecheyanu.[5]) Not only do we say Shehecheyanu at a pidyon haben, but the Rama[6] says that if the only time the fruit will be available for one to make a Shehecheyanu is during the Three Weeks, then one should not lose the opportunity.
One common question about which there is major discussion among poskim is, as you asked, whether one may recite Shehecheyanu on Shabbat. On one hand, even though the laws of aveilut almost disappear on Shabbat, our issue here is whether one may praise the time of year. Even on Shabbat it would, apparently, be inappropriate to praise the Three Weeks. On the other hand, because of the mitzva of oneg Shabbat,[7] we do not want to refrain from adding to our enjoyment. Although there is no clear consensus on the matter, different factors can help a person decide whether it is preferable to eat the foods and make a Shehecheyanu or not.[8] One factor is how close it is to Tisha B’av. Another factor is how important eating the new fruit is for the enjoyment of Shabbat.
In general, with regard to the laws and customs of the Three Weeks, it is important to keep things in perspective. Many of the specific customs are relatively recent, and sometimes the extent to which they are binding on a given community is questionable. On one hand, the existence of customs, many of which are halachically binding, are crucial in creating a pervasive atmosphere of dampened joy. On the other hand, the heart of the mitzva is the atmosphere itself.