My family is not frum, but they do not eat non-kosher meat at home, although they do eat other not-certified-kosher products, including cheese. They don't mix meat and milk, but they do wash meat and milk dishes together. Although I have kept kosher for a while now, this is the first time (since I started keeping kosher) that I will be living at home for a significant period of time. I have my own glass dishes and my own cutlery, but I have been eating food made in my family's pots/pans, etc. Are there ANY leniencies that a person in this situation, can rely on to eat this food or use their dishes?
There might be some wiggle room for you regarding nosein ta'am bar nosein ta'am. Keep in mind that these are intricate and complicated halachos, and subtle differences in the particulars of a case can have a profound impact on the applicable halacha. As can never be reiterated enough with respect to issur v'heter, CYLOR.
Consult the Kitzur Shulchan Aruch and the Ben Ish HAi. Among Sephardic authorities glass dishes are not porous. Therefore, they cannot become nonkosher by eating meat or dairy on them. Though the dishes must be eaten separately. They must be taken to a Mikvah for purification. Metal pots and pans have to be made kosher through both a Mikvah and boiling between use. Better to buy your own pot pan and dishes. You must also buy your own silverware, however, these can also be boiled. A person my keep kosher by eating cold foods and being a vegan or vegatarian. This is not only easier when around non-Jews or Jews that do not keep kosher, but healthier. Anyother questions, please ask.
You mentioned dishes being washed together. Hacham Ovadia Yosef writes in Yabia Omer (Y"D 10:4) that you may wash meat and milk dishes together in a dishwasher as long as the first rinse contains soap (and not just hot water). Presumably, this would also work for non-kosher dishes because the laws of disqualified food (such as if it's mixed with soap) apply to non-kosher food too, not just milk and meat.
One possibility: see the opinion of R. Raphael Saffra cited in Shoshi's answer about the Tablet-K hechsher, that cheese made without real rennet is permissible. While this opinion isn't generally accepted (and indeed that's why, as mentioned in that answer and several other ones, neither is that hechsher), perhaps that might give you halachic wiggle room to at least eat other kinds of food made in your family's pots and pans, even if not the cheese dishes themselves. (As you've already mentioned in a comment on the question, this is of course something about which to consult with your rabbi.)
You mention in that comment also about dishes that haven't been used in 24 hours. Provided that there is no visible food residue left on them, then indeed halachah considers that the taste of the food molecules absorbed into the dish is pagum ("impaired") and doesn't have the same capability as before to render food treif. This can get pretty complicated, though, and I don't know whether washing the dishes during this time (in hot water, naturally) resets the 24-hour clock; but again, that's something for a rav to decide.
In any case, I wish you hatzlachah, and an easy time of it, in your Jewish journey!