While you can buy tickets for each Łazienki attraction individually, you can also purchase a one-day ticket (40/25zł) which offers single-access to a chunk of the attractions (the Botanical Garden is not among them).
Łazienki, up until very recently, was further evidence of the fact that many Central Europeans have never quite grasped the idea of what parks are actually for. With superbly kept grass at every turn, perfect for picnics, pick up games of cricket, softball, football or whatever else it is people get up to in parks in the western world, Łazienki once took a very stern ‘look but don’t touch’ attitude when it came to its lawns. That said, the park now allows you to have a picnic on all patches of grass in the park. Finally, progress!