По дороге к метро иди осторожнее: после дождя на улице скользко.

По дороге is a set phrase meaning along the road / on the way.
Grammatically, по here takes the dative case, so дорога → дороге (dative singular). It’s very commonly used to mean “while going somewhere”: по дороге домой, по дороге на работу, etc.

К + dative means toward / to (the direction of) something: you’re going in the direction of the metro station.
В метро would mean into the metro / in the metro (subway system), i.e., you’re already entering it or inside it.

Метро is an indeclinable noun in Russian (it doesn’t change by case).
Even though к normally requires the dative, you still say к метро, not к метру.

Yes. Иди is the imperative (command) form of идти (to go, on foot).
It’s addressing one person informally (ты). For polite/plural you’d use идите.

Осторожно = “carefully” (plain adverb).
Осторожнее is the comparative form: “more carefully / more cautiously.” In context it often means “be extra careful.”

It functions as an adverb modifying the verb иди (“go more carefully”).
It’s the comparative form built from осторожный → осторожно → осторожнее.

The colon introduces an explanation or reason:
Do X: because Y.
It’s a common written style in Russian. You could also say:

• … иди осторожнее, потому что после дождя на улице скользко.
  The colon version is shorter and a bit more “note-like.”

После usually takes the genitive case, so дождь → дождя (genitive singular).
После дождя literally means “after the rain.”

Russian typically uses на with open public spaces/surfaces:

• на улице, на площади, на мосту
  В is more common for enclosed spaces or “inside” something:
• в доме, в магазине, в парке (park is treated as an area you enter)

Скользко is a category of state word (often called a “predicative”), used in impersonal sentences to describe conditions:

• Здесь холодно. (It’s cold here.)
• На улице темно. (It’s dark outside.)
  Russian often omits “to be” in the present tense, so the idea is “it is slippery” without an explicit verb.