If you want to visit Sala Hipòstila within Park Güell, purchase tickets in advance online. It’s best to buy them a few days prior as they tend to sell out.
Most of Park Güell is free to the public. You will only need tickets for the Greek Theatre and Sala Hipòstila (the famous mosaic benches from which you’ll have a wonderful view over the city).
Late afternoon has the best lighting.
I would avoid the height of the summer as it will be very hot, but any other time of the year will be nice.
Most of the park is free. Skip Sala Hipòstila if you are on a budget.