Ich hänge die Jacke an den Haken, bevor ich ins Bad gehe.

Because the verb hängen is conjugated for ich (1st person singular) in the present tense:

• ich hänge
• du hängst
• er/sie/es hängt
  So hänge is the correct form with ich.

Because an is a “two-way preposition” (Wechselpräposition): it can take

• accusative for movement/direction (putting something somewhere), and
• dative for location/static position (being somewhere).

Here you’re moving the jacket onto the hook (direction), so it’s accusative:

• an den Haken = onto the hook

If the jacket is already hanging there (location), you’d use dative:

• Die Jacke hängt am Haken. (am = an dem)

They are different cases of the masculine noun der Haken:

• der Haken = nominative (subject)
• den Haken = accusative (direct object / direction after a two-way preposition)
• dem Haken = dative (indirect object / location after a two-way preposition)

In this sentence, den Haken is accusative because an expresses placement/movement.

Because bevor introduces a subordinate clause, and in German subordinate clauses typically have verb-final word order:

• ..., bevor ich ins Bad gehe. (verb gehe at the end)

In the main clause, the conjugated verb is usually in second position:

• Ich hänge ... (verb hänge is 2nd)

German normally includes the subject in each finite clause. So:

• main clause: Ich hänge ...
• subordinate clause: bevor ich ... gehe

In English, you sometimes drop it in casual speech (before going to the bathroom), but German usually needs a full clause if you use bevor + clause.

ins is simply the contraction of in das:

• ins Bad = in das Bad

Also, in is another two-way preposition:

• ins Bad (accusative) = into the bathroom (movement)
• im Bad (dative, im = in dem) = in the bathroom (location)

Both are possible, but they’re used a bit differently:

• Ich hänge die Jacke an den Haken. is natural and common when you specify the place with an + accusative.
• Ich hänge die Jacke auf. uses the separable verb aufhängen and can work even without saying where (though you still can add it).

So hängen + an den Haken already conveys the “hang up” idea clearly.