As janpona points out, 'lawa' is the usual word for "drive" applied to cars and the like,"control, command, direct" (derivative -- very -- from its basic meaning of "head"). Driving away animals is most likely 'kama weka e soweli' "cause the animals to be distant"; in any case, 'lawa' usually plays no role in it, nor 'tawa neither. 'lawa' might enter in if the driver were the herdsman, 'jan pi lawa soweli (or, probably, 'mani' "herd animal")' for he would indeed lawa weka e mani ona.
Alas, janpona's language, while looking like tp, often is not, for 'noka' is not standard tp for "go", except, possibly, for "walk". "We drive toward home" is just 'mi mute li tawa tomo'. "We go toward home (the "drive" is either contextual or added on 'kepeken tomo tawa'). 'tawa' is basically the preposition "toward" which picks up the meaning "move, go" inferentially when used in verblike situations. As a preposition, its object -- the goal or destination -- is simply expressed by a noun phrase immediately after the preposition (or its modifiers), not as a direct object (with 'e') like a verb. 'taw' can be made into a transitive verb as a causative, with the direct object then the thing the subject causes to move (typically, to some place). So 'mi tawa a tomo tawa' means "I moved the car", with driving it a possibility but not one suggested by the form, whichis parallel to "I move the box" 'mi tawa e poki'. And, just as I can move a box home 'mi tawa tomo e poki', I can move a car home 'mi tawa tomo e tomo tawa'. So, once you get the 'tawa' into a sentence for the going, you do not have to repeat it for the destination, although you may want to if the sentence gest too complicated otherwise: 'mi tawa tomo pi mama mama mi e poki' "I took the box to my grandmother's place" might be easier to read as 'mi tawa e poki tawa tomo pi mama mama mi'.
'weka' is an adjective meaning "away, distant", so again, a transitive form would be a causative meaning "cause to be distant", which clearly covers driving away. ('weka' is also used reflexively, so that 'mi weka' means "I go away"-- this may be a regular feature, since it works for 'tawa', too).