Buses are most frequent in July and August, and services thin out on weekends. Carefully check schedules at https://cp.hnonline.sk.
There are up to five daily direct buses from Poprad to Hrabušice (€1.30, 45 minutes) and Dedinky (€2.10, 1¼ hours). Alternatively, change buses in Spišský Štvrtok.
More frequent buses from Poprad reach Slovenský Raj's transport hub Spisška Nová Ves (€1.70, 45 minutes, at least hourly Monday to Friday and every one to two hours Saturday and Sunday). There are a few more services if you don't mind a change of buses in Levoča (almost doubling the journey time).
If you arrive into Spisška Nová Ves and need to reach the park, it's easiest to hike, cycle or arrange a transfer through your accommodation. If you don't mind waiting around, infrequent daily buses run between Spišska Nová Ves and Hrabušice (for Podlesok; €1.10, 30 minutes), Čingov (€0.60, 15 minutes) and Dedinky (€2.60, 80 minutes).
Trains run from Spišska Nová Ves to Poprad (€1.55, 20 minutes, at least hourly) and Košice (€4, one to 1½ hours, every one to two hours).