Take the bus to Bratislava main bus station with Slovak Lines, departing 30 minutes after the hour for the most part of the day. Tickets can be purchased with a credit card from the driver, or online here . Alternatively, you can take bus operated by Flixbus, order tickets here.
After arriving at Bratislava main bus station, take the trolleybus no. 40 to the main railway station (the final stop). Tickets can be purchased by tapping a credit card on the bottom part of one of the ticket validator machines inside the trolleybus (it costs €1.20). Or, you can buy a standard ticket from a kiosk, or ticket machine (?) valid for 60 minutes regular ride for €1.10.
From Bratislava main train station, take the train to Trenčianska Teplá, leaving hourly 5 mins before the hour: here is a regular schedule. Ticket office is above stairs on the right-hand side to the entrance of the main station building (coming from bus stop).
From Trenčianska Teplá, take either a local bus to Trenčianske Teplice (tickets from the driver) or a taxi, which should be under €10.
Take the bus no. 61 to Bratislava main railway station (tickets as above), which is also the final stop. Then continue as above.