Kyoto is the former capital of Japan and, according to many sources, the most beautiful city in the whole country. Despite this, it is not necessarily very easy to navigate so you will generally have to dig deeper than usual to find the most beautiful and spiritual places, but it is definitely worth it. The city is teeming with culture, and many, many backpackers visit Kyoto hostels every year to see the magnificent temples and parks. Kyoto is home to over 1.4 million people.
There are many ways to reach Kyoto. Flying is a common option (although the nearest airport is Osaka - Kansai), the Shinkansen (the famous bullet train) is another. There are also more conventional trains available, and the city is also served by a comprehensive bus network. Overnight buses are probably the cheapest budget option, if you are traveling from further away, like Tokyo.
Although Japan is often referred to as an expensive country to visit, it does not necessarily have to be. Hostels in Kyoto, for example, are of good quality and the prices are reasonable. If you shoot for the cheapest option, go for the dorm beds, which will set you back about 15-20 US dollars per night. Depending on the season, Kyoto hostels are often fully booked, so be prepared to reserve a space in advance.
When traveling inside the city, the public transport system is your best option. Although biking is generally a great option, bear in mind that it is a huge city so if you plan to bike from your Kyoto hostel try to avoid the high traffic areas and you should be fine.