Learning Cyrillic (see Russian Alphabet) well from the start prevents a specific, common trap: silently mispronouncing words based on their Latin-alphabet "false friends."
Study В (v), Н (n), Р (r), С (s), and similar letters as a dedicated group before anything else — these are the letters most likely to cause silent mispronunciation if you unconsciously read them as their Latin look-alikes.
Letters with no Latin equivalent (Ф, Щ, Ы, and others) have no false-friend trap to worry about — they're often easier precisely because there's no misleading familiarity to overcome.
The hard sign and soft sign don't represent sounds of their own — learn them explicitly as modifiers of the preceding consonant's hardness/softness, rather than trying to find a sound to attach to them directly.
Because Russian cursive letterforms can look substantially different from print, introducing cursive too early risks confusing two skills that are more efficiently learned in sequence.
See also the reference page: Russian Alphabet.
Silently mispronouncing letters that look like familiar Latin letters but represent different sounds (В, Н, Р, С) — isolating and studying these specifically first prevents the trap.
Often easier, in fact — since there's no misleading Latin look-alike to overcome, letters like Ф, Щ, and Ы can be learned without the false-friend confusion the familiar-looking letters cause.
Not sounds of their own — they modify whether the preceding consonant is "hard" or "soft" (palatalized), so they're best learned explicitly as modifiers rather than sounds to pronounce directly.
It's more effective to delay cursive until print recognition feels solid, since Russian cursive can look substantially different from print and risks confusing two skills better learned in sequence.