Learn Danish: Hvad kan du tegne?

Because it follows the modal verb kan (can). In Danish, just like in English, modal verbs are always followed by the infinitive (the dictionary form) of the main verb.

Should I use at (to) before tegne?

No, you leave it out. When using a modal verb like kan, you use the bare infinitive. It works exactly like English: we say "can draw", not "can to draw".

Why is the word order kan du and not du kan?

This is because Danish is a V2 (verb-second) language. The question word Hvad takes up the first position, so the verb kan must come in the second position. This pushes the subject du right after the verb.

How do you pronounce Hvad?

In the word Hvad, both the h and the d are completely silent! When speaking, it is pronounced simply as "va".