We climbed up to Arthur's Seat last Sept. I thought that i had it figured out to save us some steps but I was WRONG... I'll tell that at the end. So i thought i'd write a note for First Timers because i couldn't find anything when we were planning our trip.
In our opinion the Best and easiest route is to to start at Holyrood Palace. Catch a bus(35?) there and then head off. Very quickly you'll be able to enjoy the ruins of St. Anthony's Chapel. It will be a steady climb and allow lots of time to do the last bit. We wore hiking boots and knew we'd be doing a climb but I really was not prepared for how gnarly the rocks are to get to the top. and lets face it once you get that far...you need to get to the very top. 360 degree views. It's spectacular!!
Wear proper footwear (no flipflops) and just take your time to pick a path slowly and try not to turn an ankle.
And I would not say it's a 20 min hike like I had read. Probably more like 2 hours. But you could go down the hill towards Dr. Neils Garden and catch Bus #12 back into Edinburgh. Check routes.
We thought it would be easy to access from Duddingston Church. I didn't take into account that it is all uphill from there. We rode a bus to the Holyrood High School Stop. We took a quick look around Dr. Neil's Garden. it looks to be lovely from what we saw. Unfortunately the church(Duddingston Kirk) was closed even though my research said it would be open until 5. Which is the other reason we went that way. :(
Hope this is helpful to anyone headed to Arthur's Seat for the 1st time.