I think there are basically two options: One is to stay in the North of the park. This is the somewhat more touristy part, where the museums and restaurant are located. You could take a white bike, or hike from here to one of the wildlife observation points that are on the official map.
The other option is to explore the South part of the park. There are no services here other than the occasional picnic table, and one teahouse near the South entrance. Finding a shelter in case of rain is going to be a challenge, so bring proper clothing. Especially in October, you are not going to see a lot of other people here, giving you a better chance of spotting animals.
Since you said you were looking for trails, and more interested in nature, an option would be to hike from either Otterlo or Hoenderloo to the South entrance at Schaarsbergen, or the other way round. Both are part of the ‘Veluwe Zwerfpad’ marked trail, and will give you nice views over the park. Details can be found at waymarkedtrails.org, look for section 13 (~15 km), or section 15 (~19 km). Schaarsbergen is reachable by bus nr 9 from Arnhem.
IIRC, both sections are unpaved almost the entire way, so you wouldn't be able to take a white bike, unless you take it to a place where the trail crosses the bicycle path, and hike from there.
Alternatively, stay North and hike from Hoenderlo to Otterloo (section 14, ~12 km). With regards to nature, I personally found this section to be less impressive than the ones above (still nice enough though), but it will give you the opportunity to visit the Kröller-Müller museum, or even just the freely accessible sculpture garden, or Jachthuis Sint Hubertus, a 100 years old country house. All are highly recommended.
Whether its worth going even if some rain is expected is a matter of personal preference. If its raining all day, you'll probably be miserable by the end of the day no matter where you go in the park. Animals will probably be hiding too. You may want to check the forecast before your departure.
