Japan is jam packed with places to visit, if you look hard enough something old and fantastic is probably no more than a twenty minute leisurely bike ride away. Kyoto is a perfect example of this, almost to the extreme degree. But the lesser known, Nara is just as full of beautiful, ancient places, most much much older than your standard Kyoto fare.
Making it to Nara is relatively easy if you're already visiting another place in Kansai (Osaka, Kyoto, Kobe, etc.). It's around an hour's train ride from Osaka (probably faster if you pay the extra cost of hopping on an express) and maybe a bit more from Kyoto. I started from KintetsuNara on the Kintetsu Line, but starting from Nara Station on the JR Kansai Main Line doesn't really put you that far off the mark.
The walk through Nara Park up past Kofuku-ji (which is partly under construction, boo!) is nice on a good day, but beware – once you've purchased wafers to feed the deer in the park, good luck having any peace. Pushy little buggers!
Todai-ji is a must see, as it's complex houses the world's largest bronze statue of Bhudda, and the entire complex, along with seven other sites in the city, is a UNESCO World Heritage Site. (Don't know what a World Heritage Site is? Do yourself a favor before coming to Japan and look it up. Japan is obsessed with them and very, very proud of the sites in Japan that have received the designation.) You can see quite a bit of the complex before having the pay the entrance fee to the main hall (Y500), but it's worth the coin to get and wander around.
I recommend buying a bento and bringing it with you through Todai-ji and the shrine's near by, and have a nice picnic lunch in one of the lovely open shelters in Nara Park. I unfortunately got hungry when I reached Tamukeyama Hachiman-gu Shrine and had to head back to find food, after which the weather became absolutely miserable and I had to call it a day. Major bummer. Don't make that mistake.