Learn Russian: Я тоже изучаю русский язык.

In Russian, the verb you use for study changes depending on what you mean. We use изучать (here conjugated as изучаю) when we mention the specific subject we are studying. The verb учиться is used to say where or how you study, like studying at a university.

I thought the object of a verb takes the accusative case. Why does русский язык look like the dictionary form?

It actually is in the accusative case! Because язык is a masculine, inanimate noun (a non-living thing), its accusative form looks exactly the same as its dictionary form. The adjective русский copies this rule, so it doesn't change spelling either.

Do I have to include the word язык? In English we just say 'I study Russian'.

Yes, it is much better to include it. If you just say Я изучаю русский, it sounds incomplete and a native speaker might think you mean you are studying a Russian person. Adding язык makes it clear you are talking about the language.

Can I put тоже at the end of the sentence, like the English word 'too'?

Russian word order is flexible, but тоже usually goes right after the subject it refers to. Since you are saying that you also do this, it sounds most natural right after Я. Putting it at the very end would sound less natural to a Russian ear.