Travelling from Edinburgh to Kilmarnock using public transport is still fairly straightforward although there arent any direct services. Scotrail services can get you there in around 2hours with up to 2 changes. Buses are also available via Buchanan Bus Station which are also just over 2 hours in duration. You can view different routes either directly through Scotrail or via Rome2Rio website which will show you various routes to Kilmarnock.