Quick answer
You travel на по́езде, in ваго́н number so-and-so, on ме́сто number so-and-so. Купе́ never declines.
The train
The journey
On board
Which class to book
Купе́ is a closed four-berth compartment. Плацка́рт is an open carriage of 54 berths, cheaper and famously sociable. СВ has two berths and costs roughly double a купе́. For a first long Russian train journey, купе́ is the easy choice and плацка́рт the memorable one.
The etiquette
The lower berth doubles as everybody’s seat during the day, so its owner shares it. Food is offered round and refusing twice is normal. The attendant brings бельё and wakes you before your stop if you ask. And Моско́вское вре́мя — Moscow time — is used on long-distance timetables regardless of where you are.
Typical mistakes
в купе́е
в купе́
купе́ is indeclinable, like метро́.
Я е́ду в по́езде — meaning by train
Я е́ду на по́езде.
Means of transport takes на.
Check yourself
What is плацка́рт?