The best way to get to Yala is by private transport as it is the fastest and most direct way. You can head to either Kataragama or Tissamaharama which are the two largest towns near the park that offer accommodation and safari jeep hires. The journey from Colombo by car is about 5 hours. Tissamaharama is slightly closer (by about 15-20 minutes) than Kataragama, but further away from the park, so it will mean an earlier wake up time (by about half hour - 45 minutes) for your morning safari.
If you prefer to take public transport, you can also catch a bus from Colombo to Kataragama, which will take about 8 and a half hours.