Looking for the best driving route from Krakow to Bratislava. Are roads easy to follow?Are road signs available? Are the roads over the mountains paved? And should we expect snow at the mountain passes in early October? Thanks

First, I would love to do that trip! Second, snow would be quite unlikely. Are you familiar with driving in those countries? It is quite doable, but nothing like driving in the US. Roads are narrower and there are more hazards and less warning of dropoffs, potholes, etc. Go slow and cautiously on secondary roads. Other drivers may not always do what you expect. Also, do you speak either language? Signs on the main highway are pretty good, other than that, dodgy. The best tip is to have a good map (or three) and to KNOW the names of the next three towns/cities on your route and follow those signs. The second half of your trip - Zilina to Bratislava, is the easiest, it is all super highway. Anyway it is the best known to me, and the signs are clear. It is a nearly straight
road from Zilina to Bratislava, and is a first class limited access highway, similar to but not exactly the same as an Autobahn or US Interstate. Traffic will be going 80-150+kph. That means some of each speed, which means you REALLY have to pay attention at all times. If you decide to go into the Left Lane, take a LONG look behind you first, some of those faster cars will be on you in an instant, possibly with fatal results. Also, in Slovakia, You will need to buy a highway pass at the First gas station you come to, and stick it on the inside of your windshield. fines for failure are heavy.
From Krakow I am not so familiar, but it looks at least feasible to go Myslenice to Rabka, across the border to Valaska Dubova / Ruzomberok and from there I know the road is quite good to Martin, and then Zilina, though these will include some secondary highways where a TOP speed of 70-80 kph is a reasonable maximum, as these roads often have no white line on the right side and sometimes not even a center line.
I did drive a little in southern Poland and the roads were quite ok, recognizing the above characteristics. I don't know about highway stickers in Poland. I just re-read your question, and I found it quite easy to get off the proper road, you really have to pay attention, but for the most part they are fine. If you are european, probably you know most of these things already, and I apologize for overdoing it, but most people from US would need this info!