I can't pin down how "relaxed" you want to be when touristing, so not sure how much to put on your plate for a single, not-so-long, day, but
a) I've never spent less than 3 days in Kyoto when I have gone there. I was there for a month 2 summers ago. Saying, I'm not sure what you should / could see in 6 hrs after arriving at the main train station. I also only used a taxi in Kyoto twice in a total of 6.5 wks there. Not sure how much time they will save. So,
#1 Kyoto thought = the Imperial Palace & surrounding park is a top site & it is not far N/W of the train stn. Taxi to it. I wandered around there 3x in the month 2 summers ago.
This is the only 1 in Kyoto where I think you can relax and not worry about time. There are other things to see not far east of the main stn, but I think my #2 dominates them easily.
#2 Kyoto thought = if you can figure out how to do it in the time you have, my top destination would be the far N/W corner where Kinkaku-ji, Ryoan-ji, & Ninna-ji are all within walking distance of each other. You have undoubtedly seen pics of the last 2 & maybe all 3 (iirc, Ninna-ji is where Scarlet J walked around in Lost in Translation & Ryoan-ji is the "place with the rocks" (LOL). PRoblem is time. You may not walk as quickly as we do and depending on when you finally get there, you may need to taxi between them. Seeing all 3 is a WOW day, but you need to do a schedule.
Which train to Kyoto? arrive what time? maybe it is AM traffic hour, though you'd be surprised how small of a town Kyoto actually is, not in SqMi, but in hustle/bustle)
Taxi faster to get out N/W or get on an express train going due west, then get off at the stop due south of Ninna-ji? If train, then taxi north to those 3.
#3 Kyoto thought = pretty close to main stn, to the east is a mountain with shrines for a fox-god. whole place is named Fushimi Inari-taisha. There are things to see at the bottom of the mtn but the deal is that in those days the monks who built this stuff were often seen as a threat by the shoguns (government, as it were). So monks built a lot of this stuff on the outskirts and up on the mtns. Kyoto is almost 300 deg surrounded by mtns iirc. google Fushimi Inari, but realize to get the real view and real knowledge you should climb halfway up a mtn. If you do, you will realize there are many 80 yr old Japanese ladies climbing with you.
You can see the letter "K" dominates the naming around Kyoto, so if I mixed up which name goes with which ... oops. Google can get it right
#4 Kyoto thought = another mtn one which is awesome but not on std "tourist" lists. In the northern mtns, you can take a scenic train up a beautiful valley to a mtn town called either Kuramasou or Kifune. You'd take a train back to Kyoto from the other one. In between, you can hike over a small mtn called Kuramadera with shrine of same name. same story as Fushimi Inari. Monks hiding in the mtns basically owned this mtn and built trails and shrines all over it.
#4 is only if you're endurance-minded and hiking over a small mtn while seeing great stuff will add to the experience, but TIME will be big factor on this one.
I could do all of these in 1 day from where I stayed in Kyoto without getting up early and using local buses to get around, but it was summer with long days and no need to be back in Osaka at 4 pm.
and in #4, you could get lost and there will be much much less chance to find solid English-speakers around
anyway those are the names to google & that was a brain dump for me to read before I go back to Kyoto again