Kyoto's a very compact city, hemmed in on three sides by forested hills, so it doesn't take too long to get out into the hills, though most of them are covered with established secondary growth or plantations - extensive conifer and localised bamboo. It's only in temple and shrine precincts that old-growth trees remain. While being able to get out of the city with ease can be a big plus, visitors would be as well to stick to the city parks, the Kamogawa and two or three good spots abutting the hills. Unlike the large parks in the vast urban sprawl of Osaka, which routinely concentrate an amazing range of migrants, the close proximity of the forests around Kyoto probably lessens the pull of its parks, nevertheless they're the best bet to see the commoner birds in a short time.
Some migrants, flycatchers such as Grey-streaked, Asian Brown and Narcissus are common enough, reflecting their status generally. However I've never seen some scarce migrants such as Paradise Flycatcher in the city, they do occur but but seem easier in the surrounding hills provided you have the time to put in. Likewise Siberian Blue Robin and Japanese Robin, views can be great if you're lucky enough that your visit coincides with a migrant in the park but there are more of them in the surrounding hills. Similarly, in winter there's a greater range of species on the surrounding hills but not only can they take longer to find depending on the species but they tend not to give the great views wintering park birds very often do; Red-flanked Bluetail and Pale Thrush would be obvious examples of birds that can be surprisingly confiding in the city but are more difficult to see well in the forests.
There are two important parks in the city, the botanical gardens and the Imperial Palace, both easily reached by subway. Years ago I was told the Palace was better during migration and the gardens in winter. There's some truth in that, in winter at least, but I'm not sure all the migrants know where they're supposed to be.
Imperial Palace Park
The park is basically a large area of cropped grass covered mainly with mature pines and maples which in itself isn't terribly attractive except to Dusky Thrushes and Olive-backed Pipits in winter. However it's fringed with huge broadleaf trees, the width of the band varies but the southern, northern and eastern edges are best. The north east section is set aside as a "natural" area and the surrounds of ponds in the south east corner (just north of the tennis courts) and mid-way along the north side can be excellent.
Apart from being good for migrants, it's a great place to see breeding Brown Hawk Owls, the pair towards the north east corner are easiest to see and the nest boxes set out for them are easy to find.
It can be reached from the Karasuma line of the subway from Marutamachi station (exit 1) at the south west corner or Imadegawa station (exit 3) at the north west corner.
The botanical gardens has the same common resident species as the Imperial Palace Park, Japanese Tit, Varied Tit, Japanese Pygmy Woodpecker and so on, it's smaller and much easier to cover thoroughly. It's well worth a visit in winter or during migration. Flocks of Japanese Grosbeak are often very approachable here in winter, as are Pale Thrushes and even Red-flanked Bluetail. In waxwing years there's a good chance of seeing Japanese in the general area around the gardens.
The gardens are located next to the Kamogawa and a walk along the river is also an excellent way to see dabbling ducks, the local wagtails and possibly Long-billed Plover if you're lucky. The Uji River and Katsura River are much better for the latter.
The easiest way to reach the gardens if not walking along the river is Kitayama station (exit 3) on the Karasuma subway line.