Arguably the best way to explore Asuka is by bike. There are numerous bike rentals outside Asuka Station. The weekend prices are 1000 yen for a normal bike (weekdays are cheaper. Some rentals offer motor assist cycles for a higher price).
The bike renters should give you a map of the area with the sites listed in Japanese, although I would recommend printing a map ahead of time and marking the places you want to go to make it easier on yourself and make sure you don't miss anything you want to see and you can establish a sensible route (for me, I used both the maps I brought and the ones given to me). The area has a ton of sites, but most likely not all of them will interest you. Still, it's nice to know that they're there and stop along the way to see some of the minor sites.
If you start reasonably early and travel at a moderate pace, you should be able to see many sites in a day; probably everywhere you'd want to visit. For example, you can visit the Takamatsuzuka Kofun (burial mound), the Kameishi (turtle rock), Ishibutai, Oka-dera, Tachibana Temple, Asuka-dera, the Manyo Museum, and Asuka Museum and still be able to stop along the way to see a few other places.
Don't let the rural atmosphere fool you; Asuka is very easy to reach by train. There are a few stations in the town but Asuka Station is the best entry point. The station is served by Kintetsu Railway.
From Kyoto, you can take the Kintetsu Line to Kashihara Jingu-mae and transfer there to the Yoshino Line to reach Asuka Station. If you go by express trains, it shouldn't take more than 1.5 hours.
From Nara, make sure to use Kintetsu Nara Station and go to Yamato Saidaiji Station. There you can transfer to go south to Kashihara Jingu-mae where you will transfer to the Yoshino Line to reach Asuka Station.
From Osaka, if you are on the Loop Line go to Tennoji Station. From there you can walk to Osakaabenobashi Station and go directly to Asuka Station via the Minami-Osaka Line (which becomes the Yoshino Line once it leaves Osaka).