There are three ways to get to Penang from Kuala Lumpur: flying is the best way, but you can also take a train, or take a bus. Bus is actually very popular for Malaysians
Flight (best way):
Flying is the best way to travel because it's usually pretty cheap and fast. I usually pay around $20-$30 USD for a one-way flight. Return flight is around $50 USD
Flight itself takes 1 hour. Total trip time should be 3-4 hours. There are many flights everyday
12Go or Skyscanner are both good websites for booking flights in Malaysia
From Kuala Lumpur Sentral, you need to take the train to Butterworth Station and transfer to a 20-minute ferry to George Town from the docks next to the train station
There are 4 trains everyday to Penang Butterworth Station from KL at: 11:25, 14:40, 20:40, 21:50
Train takes 4 hours and 15 minutes. Total trip time is 6-7 hours
Ticket is around $20USD
Best place to book is either the KTM website (Malaysian's main train company) or 12Go
Once you get to the Butterworth train station, take the free shuttle to the ferry terminal 400 meters away
Ferry runs between 5:20 to 0:10 (10 minutes past midnight), about 2 ships per hour
Fare is 30 cents USD, paid by cash at the terminal
There are many daily buses going to Penang from KL. Most of them depart from TBS station, some also leave from the bus terminal at old KTM train station
First bus of the day is usually at 9:00
Bus is very cheap, about $10 USD each way
Bus takes 5-7 hours depending on which bus you take. Total trip time is 7-9 hours
Buses arrive at either Butterworth Bus Terminal (next to the ferry terminal and across the street from the train station), or Sungai Nibong bus station (on the Penang island itself, halfway between the airport and George Town). My recommendation is to take the bus to Butterworth, as it's a short 20-minute ferry ride away from George Town. From Sungai Nibong you need to either take a taxi or take a 1-hour bus to get to George Town
When you get to Butterworth, walk over to the Pangkalan Sultan Abdul Halim ferry terminal 200 meters from the bus terminal. Then take the 30-cent-USD public ferry to George Town. It's the same ferry you'd take if you take a train to Butterworth. Check the official site for its current ferry schedule and fare. The ride only takes 20 minutes and there are about 2 ships every hour.
12Go is the best website for looking up the schedule and booking buses