If for some reason I am held up doing something until after the time that mincha must be done by and I have not davened mincha, is it better to daven mincha after the time it is supposed to be done by or should I just not do it? And also with shachris and other prayers, is it better to daven late than never?
The Mishna Berura rules that one should not start mincha if they will not finish Shemoneh Esreh before shkia (sunset), but many rely on Rabenu Tam's time and daven considerably later.
I was told by my rav to omit tachanun if davening later than allowed by the M"B.
Shachris can technically be davened until chatzos (halachic midday) but it really ought to be davened by four halachic hours into the day - the "latest Shemoneh Esreh" time on sites like myzmanim.com.
Maariv likewise should ideally be said by halachic midnight, but can be said until dawn.
These are all the most lenient positions, and one should strongly prioritize davening at the correct time, only relying on these times in great need. Many authorities rule much more stringently. For practical halacha, consult your local orthodox rav.
Yoel's answer describes until when one can pray. I'll just add that if one missed the right time due to circumstances beyond his control then he can usually make up the missed amida (sh'mone esre) by reciting an extra amida after his next one; e.g., if he missed mincha he can say two amidos at maariv. (This does not apply to a missed musaf (and I'm not sure about n'ila).) If, e.g., mincha was Shabas's and maariv is rosh chodesh's, he recites two weekday maariv amidos and includes "Ata chonatanu" in the first, which is for maariv, and "Yaale v'yavo" in both, since it's rosh chodesh. (All this is in Mishna B'rura 108 passim.)