Feminine nouns are almost always regular. Well, feminine nouns ending in consonants may have plural nominative/accusative -e (paproć-paprocie) or -i (nić-nici), but that’s a rare exception. Also ręka-ręce.
Neuter nouns are perhaps even more regular. As long as you don’t confuse the -mię nouns like imię/imiona with the -ię nouns like szczenię/szczenięta. Also oko-oczy, ucho-uszy.
Masculine nouns can be a bit trickier in a few cases, with singular genitive -a or -u, and singular dative -owi or -u. There may also be some irregularities in the plural nominative and genitive (mostly in nouns referring to humans, like oficer-oficerowie/oficerzy). There are also animacy distinctions in the accusative, which is a separate topic.
That still leaves many cases being very regular:
The plural dative is always -om.
The plural instrumental is always -ami except half a dozen nouns with -mi (ludźmi, braćmi, dziećmi, końmi, dłońmi, liśćmi), and three body parts which allow both -ami and the old dual form -oma/-yma (rękoma/rękami, oczyma/oczami, uszyma/uszami).
The plural locative is always -ach, if we ignore a few country names (Niemczech, Węgrzech, Włoszech).
The singular instrumental is always -em/-ą.
The singular locative and vocative also follow strict rules, with only a few exceptions (domu, synu, Boże).
There are also a few Latin neuter words like akwarium which are for some reason indeclinable in the singular but not in the plural (akwaria), and even use a seemingly masculine plural genitive ending (akwariów).
You generally only need to know the infinitive and first and second person singular of a verb (e.g. nieść-niosę-niesiesz), there’s only half a dozen irregular verbs where you might need to know the third person plural. The past tense is usually simple, but you have to keep in mind some vowel alternations (niósł-niosła-nieśli). Also the imperative might occasionally be slightly irregular.
native, 
C1-B2, 
B2-B1, 
B1-A2, 
A2-A1, 
A1
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