Il gatto sale sull’albero rapidamente.

• Salire is a general verb for moving upward (stairs, hills, trees, etc.).
• Arrampicarsi is a reflexive verb that implies a more effortful or technical climbing action, like rock climbing or scrambling with claws.
  You could say Il gatto si arrampica sull’albero to stress the physical effort, but sale is more natural in everyday contexts.

Sull’albero comes from combining the preposition su (meaning on or onto) with the elided article l’ (the). The contraction rules are:
• su + il → sul
• su + l’ → sull’
The apostrophe marks the dropped vowel in il before a vowel-starting noun.

Italian adverbs are flexible in placement:
• Default (sentence-final): Il gatto sale sull’albero rapidamente.
• Pre-verbal (emphasis on manner): Il gatto rapidamente sale sull’albero.
Shifting the adverb changes the nuance of focus but not the core meaning.

Most Italian adverbs derive from the feminine singular adjective plus -mente:
• rapido, rapida → rapidamente
• veloce, veloce → velocemente
Irregular adverbs include bene (from buono) and male (from cattivo), which do not follow the -mente pattern.

Yes, common alternatives are:
• velocemente (adverb from veloce)
• in fretta or alla svelta (phrases meaning in a hurry)
E.g. Il gatto sale sull’albero velocemente or Il gatto sale sull’albero in fretta.

You adjust articles and noun endings for plurality:
• Multiple cats, one tree: I gatti salgono sull’albero rapidamente.
• Multiple cats, multiple trees: I gatti salgono sugli alberi rapidamente.
Note that su + gli (plural masculine before vowels) becomes sugli, and alberi is the plural of albero.