If a mother is breastfeeding, is she obligated to fast on days such as yom kippur? If not, is there some traditional/symbolic gesture women practice, for example abstaining from their favorite foods?
Minor fast days such at Tzom Gedalya never included nursing mothers to begin with. While there is no obligation to fast, it's best to avoid delicacies or treats.
Yom Kippur is the most serious of fast days. Theoretically all adult Jews are obligated; however, there is a stronger obligation to protect life. A pregnant or nursing mother should consult their doctor and rabbi about how they (and their baby) can handle the fast.
In some circumstances where full fasting is medically contraindicated, a modified fast can involve eating/drinking up to one fluid ounce, no more often than every fifteen minutes; the halachic logic here is that this isn't as satisfying (i.e. still some sort of "affliction") as eating ad libitum. But even this may not be enough for some people's needs; in all cases, please consult with both your doctor and rabbi.
The starting point for these conversations is Mishna Yoma 8:3
עוברה שהריחה, מאכילין אותה עד שתשוב נפשה. חולה, מאכילין אותו על פי בקיים; ואם אין שם בקיים, מאכילין אותו על פי עצמו, עד שיאמר דיי.
If a pregnant woman smells food [on Yom Kippur] and craves it, feed her until she feels better. If someone is severely ill, feed them according to the [medical] experts. If there are no experts, feed them according to their own assessment, until they say they've had enough.
Breastfeeding mothers fast on Yom Kippur, (providing of course that there is no danger to the baby from his mother's fast). However, there are no other "days such as Yom Kippur" as in the question.
Regarding the other 4 Ta'anios, the mother does not fast, even though in the Shulchan Aruch (550) it is stated that Lechatchila they should fast, the custom is that there is Yeridat Hadorois and the later generations are weeker, enough to make the breastfeeding mothers exempt from the fast. Some Achromin (like the Grash Vozner) state that even today if they can they should fast.
Halachically breastfeeding is 24 months from birth (and the Maharsham quotes the Gmara that "the organs are dismantled" so that she is exempt for 24 months even if not breastfeeding) but the Achronim state that nowadays when the breastfeeding ends in practice the mother should fast.