To travel from the Thailand-Malaysia border at Padang Besar to Penang Island (Georgetown), by far the easiest way to travel is to take a cheap KTM Komuter Utara train (northern commuter).
There are regular commuter services operating on the Padang Besar to Butterworth (Penang Sentral) route and once you arrive at Butterworth Railway Station, it is a short walk over to the Penang Sentral Terminal where you can catch the ferry over to Georgetown on Penang Island (a 15 minutes boat trip).
There are no ETS train services on this route, so you have to take the KTM Komuter train.
These trains are very much like subway/tube trains with seating arrange along the side of the carriages.
To see the latest Komuter train timetable click on the following link:
You can buy your tickets from the KTM ticket office located up the stairs at Padang Besar train station.
You will have to pay in Malaysian Ringgit, so bring some with you as there is currently no money changer at the station.
Although it is possible to travel to Penang from Padang Besar, it is not really recommended as there are no direct bus services, it will take you much longer and end up costing a little more.
However, if you do want to go by bus, here is the route to take:
You will have to walk into town on the Padang Besar, Malaysia side of the border, down the hill to the small bus stop.
From here you take a MyBas bus service from Padang Besar to Kangar Bus Terminal (Terminal Bukit Lagi).
The total journey time by bus would be at least 5 hours and cost you twice as much as taking the KTM Komuter train.