The Russian word for textbook is учебник. If you want to specify the subject that a textbook deals with, you have two grammatical options. The first is to put the subject of the textbook in the genitive case after the noun:

The next option is to put the subject of the textbook in the dative case after the preposition по:

American students always pick up the Russian word класс right away because of the English word class which overlaps the Russian word in some meanings. And then immediately they try to say “I'm going to class” with the words «Я иду в класс». WRONG!!! Класс does NOT mean a class that you attend. It means the room in which a class is held. And specifically it means a classroom in a grade school or high school, not a university. Thus you could say «В класс вошла учительница» “The teacher entered the classroom.” But you can't use the word to say “I'm going to class” when you are a college student.

Класс actually has a plethora of meanings. It can mean class in the sense of first- or second-class accomodations:

The word can also mean the first years of school for a child:

The word can also be used in dividing social strata:

So how then do we say that we are going to a class? In Russian you have several slightly more specific options that are normally used. First off, if you are talking a grade school class, or a high school class, or a class outside the standard educational system, you can use the word урок lesson, which can include practical lessons like singing or playing an instrument:

If you are going to a class generically, and it's an academic adult class, and you don't need to specify the subject, then you can use the neuter plural word занятия. This is the most common way for a college or university student to say “I'm going to class”:

If you are going to a seminar or lecture class, then you can use those words, either specifying the subject or not:

The Russian word for school is школа. If a little Russian girl wanted to complain in the morning in a very grammatical fashion, she might say to Mama «Но не хочу я сегодня в школу идти!» "But I don't want to go to school today!" But actually the little girl would usually leave the 'to go' verb out of the sentence and simply say, «Не хочу я сегодня в школу!» Russian children never win that battle. A boy who goes to school is a школьник schoolboy, and школьница is a schoolgirl. Some Russian schools require the kids to wear a school uniform школьная форма.

An American college student might say "I don't work. I'm going to school these days," but in Russian школа is never used to refer to college. Instead a Russian would say «Я хожу в университет» "I go to the university," or «Я учусь в университете» "I study at the university," or «Я студент» «Я студентка» "I'm a college student."