To Bali: there are many routes to Bali, but generally all involve flying to Denpasar.
From Thailand, Malaysia or Singapore, it’s inexpensive to buy tickets online with no-frills airlines. Air Asia (www.airasia.com/th/en/home.page) has flights from Kuala Lumpur, Singapore and Bangkok to Bali. Tiger Airways (www.tigerairways.com/) flies from Singapore, etc.
Getting to Ubud from the airport: once you’re through the gates, you will walk outside and find a pre-paid taxi booth with fixed rates. Taxis/minivans to Ubud are approx. $20. The drive is about 1 to 1 1/2 hours. If you arrive in the daytime, there are also cheap shuttles that will drop you in Ubud (slow)
Once you’re in Ubud, it’s easy to get around on foot. Scooters are common and inexpensive to rent ($5 day)