"Jeździć" and "podróżować" meaning "to drive sth" or "to travel by sth" would have NARZĘDNIK = "jeździć samochodem" or "podróżować autobusem". The same applies to "pływać" if you want to say for example "to swim the breststroke/butterfly" - this would become "pływać żabką/motylkiem".
Unfortunately if you want to say "to drive/travel/swim (to) somewhere" you would have to use verb + "do" + DOPE£NIACZ = "jeździć do domu" (to drive home), "podróżować do Polski" (to travel to Poland), "pływać do brzegu" (to swim to the shore).
"Uprawiać", "lubić" and "robić" are normally followed by BIERNIK. "Tańczyć" (meaning "to dance rumba/samba") also uses BIERNIK = "tańczyć rumbę/salsę".
I'm afraid this will make your head hurt a bit more... ;)
If the verb isn't specifically another case like szukać(genitive) or martwić się(instrumental) and a direct object follows the verb, the object would be in accusative. ex. mam dziewczynę, the verb mieć always takes accusative because the noun that follows is always a direct object.
I would assume the verb to take the accusitive for the direct object unless I have learnt it to be otherwise (szukac with genitive, pozwalac with dative, etc.) It's the one to fall back on if you don't know because it's probably the most common.