Hello.
Use Yandex Maps app to Check buses and
commuter trains during your journey in Russia. For checking schedule, you also could use http://t.rasp.yandex.ru website. Tickets on commuter trains ("elektrichkas") you can buy for cash in every station or in train.
For your problem. Use bus 119 for getting from Sergiev Posad to Pereslavl Zalessky. It departs from Sergiev Posad Bus Station at 9:00 and 18:00 everyday. Bus station (Avtovokzal) situated near train station. You could buy ticket on the bus station, I think online booking is iot avialable on this destination. It will cost about 200₽.
You can use more complex variant. You can take a train 102Я Moscow - Yaroslavl (arrives to Sergiev Posad at 08:02 AM) and go to Berendeevo station (arrives at 09:01 AM). Then you wait for bus 111, which arrives at 9:50 AM. It will cost about 1000₽.
If you miss the bus, use ЯндексGo or Yandex Maps for getting taxi. You choose pay there for cash, but you need russian Phone number to get a taxi. Taxi will cost about 2500₽.
If you want from Pereslavl Zalessky to Suzdal, I found incredible route for you. You can take a bus 102 in center of Pereslavl Zalessky at 04:50 AM to Berendeevo station, then go to Alexandrov at train that arrives to Berendeevo at 06:43 AM. Journey take only 30 Minutes. Then, get a bus from Alexandrov to Vladimir (number 520) that departs at 07:45 AM. You arrive to Vladimir at 10:00 AM, and can buy a bus ticket to Suzdal. There are many buses from Vladimir to Suzdal and I think you find it fast. But you could also see Vladimir - one of most tourist cities in Russia.