Kawagoe is very easy to reach by train from Tokyo. One way to get here is by taking the Seibu Shinjuku Line from Seibu Shinjuku Station directly to Honkawagoe Station. Honkawagoe Station is also closer to the sites than Kawagoe Station. This is the fastest and cheapest way from the Shinjuku area.
If you choose to use JR, there are direct trains to Kawagoe Station from Ikebukuro Station by the JR Saikyo Line. The Tobu Tojo Line also goes to Kawagoe Station. This route will be faster and cheaper if you are in this area.
It's not difficult to get from Tokyo to Kawagoe, either on the Tobu Tojo line from Ikebukuro, Saikyo line from Shinjuku, or Kawagoe line from Omiya.
In addition, Tokyo Metro's Yurakucho line runs some through services onto the Tobu Tojo line that go to Kawagoe (some even further).
2 ways to Kawagoe, depending where is your location. If you are in Shinjuku, the straight way is by Seibu Shinjuku train from west Shinjuku station to Hon Kawagoe. It takes 1 hour (for express train) and one way costs JPY480. If you are in Ikebukuro, then take JR Tobu Tojo Line to Kawagoe station. Time about 32mins and costs is JPY450, but the main street is a bit further down.