(This is a good day to carry some snacks from the convenience store to eat picnic-style along the way or at the top of Daimonji, though there are some phenomenal restaurants too.)
It seems likely to me that you'll find lodging in the center or the southern part of the city, so it probably makes sense to start south and work your way north. The philosopher's path runs from Nanzenji in the south through Ginkakuji (i.e., the "Silver Pavilion") in the north. You can either start at Nanzenji or start further south for a longer route. This involves tacking on a few more things and starting at the Yasaka-shrine in Maruyama koen (= Maruyama park) all the way at the eastern end of Shijo (the largest East-West street running through the city of Kyoto). The stuff on the longer route can be saved for another day.
No need to spend any time at the Yasaka-shrine or in Maruyama-koen. For now, just head north-east from the entrance at the end of Shijo. These are fun spots to picnic or briskly stroll through and see the cherry blossoms. But there are greater destinations to see today! You can come back here later at night some other day to see the cherry-blossom viewing festival fair -- that's really the time to linger there and enjoy night stalls and drinking in public.
Once you've made your way to the north edge of the park (10min), you should see an incredibly gigantic gate with stairs going up into the hills. This is the Chion-in Sanmon, a national trasure. Head through it to the huge temple complex called Chion-in (founded in 1234, rebuilt in early 17th c.). The Hōjō garden on the east side is an absolute must-see! It's a recreation of a famous painting (maybe housed at Chion-in? I forget) out of shrubbery. Pay the extra fee! The Yunzen garden on the south side is also cool, but can be skipped to make time for other things if need be.
Just north of Chion-in is a smaller temple complex called Shoren-in. It has a very peaceful garden, centered around a small, green-blue pond. Good for a quiet moment. When you're done there, make your way northeast to the Nanzenji and Konchi-in temple complex.
Nanzen-ji at the southern end of the philosopher's path is known as one of the "5 great [lit. mountain] temples of kyoto." It's a pretty big temple complex, with varyingly cool parts, including an old aqueduct; several of the buildings have their own entrance fees. The Hōjō (abbot's quarters) has famous rock gardens, but my favorite part of the complex by far is Konchi-in in the southwest corner of the grounds. (I think it's an independently operating temple, actually, so it's not techincally part of Nanzen-ji). Konchi-in is not always open to the public, but it's worth checking if it is, and taking a guided tour if need be. It's stellar!
If you took the longer route, or just got a late start and want lunch, there is an awesome splurging option here: for some of the best yudofu (mild-boiled tofu -- a Kyoto speciality) try the yudofu restaurant called Okutan (Nanzenji-branch) just at the north edge of Nanzen-ji. Highly recommended. Around 3000¥ per person multi-course tofu meal.
Leave Nanzen-ji via the north exit and you'll be headed to the philosopher's path. Very soon, after passing an elementary school, you'll find Eikan-do (formally, Zenjrin-ji) to the east (on your right). This place has a lovely pond, viewable from in and outside an onlooking hall. Exit the way you came and head north along the philosopher's path.
There are three small, more neighborhood-y temples on the east side of the path, including Anryaku-ji and Hōnen-in. Give each a quick look. They aren't always open to the public, but when they are, they host things like free art exhibits. Anraku-ji is particularly pleasant in my opinion. Hōnen-in has a nice moss and sand garden in the front.
Keep heading north until the path ends near the entrance to Ginkakuji. You'll notice tourists and shops. It may look kind of hokey, but there's actually a bunch of nice establishments here. Grab some quick soba, green tea and sweets, ice-cream or etc., if you need a pick-me-up. (Note: They sell really good pickles at several shops here!)
Ginkakuji is my favorite temple in Kyoto. The variety of master gardener Sōami's very abstract gardens just blow my mind. It also represents a deeply important shift in Japanese aesthetics toward humility. It was originally meant to be covered with silver foil, mirroring the gold exterior of Kinkakji in the western hills. But through a series of dramatic events it was never finished. Instead, a tiny two-mat tea room was built here (only viewable in November, alas). This place gets SOOOOOO overrun by tourists -- especially elementary school children!! You MUST only come in the first or last hour of opening!!! In the summer that's 9–10 and 4–5, but check the spring schedule and plan around this!!
The trailhead to Daimonji is just north Ginkakuji. It's a bit hard to locate, but just keep trying until you find yourself on a nature-y, yet paved pathway. At a very leisurely pace, it takes an hour to climb up, and a little less down. The view from the summit, where the huge funeral pyre is, is amazing...you get to see *all* of Kyoto and maaaaybe even the tip of Osaka! It's nice to picnic up there a bit, or at least stop for a snack. I'd recommend trying to get to the top before sunset and watching it get dark. You'd then need a flashlight (or phone) to light the path on the way down. It could get cold in March.
For dinner before or after hiking to Daimonji, there is an amazingly well-priced and humble Kyoto style restaurant not far from the bottom of the hill that leads up to Ginkakuji and the trailhead. It's called Osai Tokoro (literally, 'vegetable place'). Their dinner sets are so yummy, mostly vegetables, and cost around $12–14 per person.
Optional add on:
My guess is that if you do this all, you will be quite tired, and could just head back to your lodging. But if you still have energy, you could walk westward along Imadegawa street toward the river (45–70 min more walking). Along the way, you'll see a lot of businesses catering to Kyoto University students, including awesome, tiny bars, restaurants, used book stores (Japanese, though), and etc. It's got good vibes in the evening, so it's fun to just walk through without even stopping anywhere. Eventually you'll hit Kamogawa (the main North-South river), and can take trains south from either Demachiyanagi-station (right at the river) or Imedegawa-station (Karasuma-line) 15 min further west.