For the second sentence it is usually understood by the listener that the class in question is your normal class or a class you are going to at that specific time. For the bank we say a/the specific bank. You can if you want specify a/the class but we often do not in colloquial speech. "I'm going to the class now/tomorrow etc" for going now we often leave out "the" because it is implied unless a specific class is stated for "now" "I'm going to the art class"

You can say either: I am going to class (not any class in particular or the particular class is not important to the converation). I am going to the class (a particular class that is known by context). This works with other nouns too... I drank milk (i.e., milk in general). I drank the milk (i.e., the particular milk we are taking about). I like girls (i.e., girls in general). I like the girls (i.e., those specific girls we are talking about). With "bank", you could say: I am going to banks (i.e., some unspecified group of banks). I am going to the banks (i.e., the specific group of banks we are talking about). You would not say "I am going to bank" because if you omit the article "the", then you are talking about more than one bank and you must use the plural form of this countable noun - "banks". Note that "bank" is also a verb, so you could say: "I am going to bank (verb) at this bank (noun). And just for fun - bank has other meanings too: "bank on" = "to count or rely on" and "bank" can also mean "river bank" = "edge of the river". So we can say crazy things like: "I bank on banking at this bank on the (river) bank." :-o

We think of "the bank" as a place, but we think of "class" as an activity. Although you are going to a particular place (a classroom, probably), to "go to class" means to go and attend a class, and it's the activity, not the place, that is important. We use this structure for some other activities. For example, you "go to violin practice." We think of "violin practice" (or football practice, etc.) as an activity, not a specific location. You can also say you are "going to dinner" with someone. "Dinner" refers to the activity of eating dinner, not the particular restaurant where you will have dinner.