If you want to make a tour with public bus, You can refer as followings:

Manjang Cave

UNESCO world heritage -
A huge cave formed by lava
(Jeju Volcanic lava tube) 

Entrance fee : 2,000 won (About US$2)
Cave Cafe

Inside of natural cave, there is a cafe where you can have green tea, bread and coffee. You can see green tea plantation also

Entrance fee : 5,000 won (About US5)
Gymnyung Maze Park

A maze park built by lots of trees. Your mission is to get out of the maze and ring a bell at the center of this park.

Entrance fee : 3,300 won (About US$3)
Sangumburi Crater
 A big crater created 12,000 years ago. You can see beatiful scenery around.
During fall, you can also see magnicficent silver grasses all over this place.
Entrance fee : 6,000 won(About US$6)
Jeju Stone Park
The large park where you can see stone statues, Jeju traditional houses, old  historic containers. There is a galleryinside also. Entrance fee : 5,000 won (About US$5)

Ecoland Train Trip
You can take a small train running through beautiful forest in Jeju. There are large grassy parks where you can take a good rest. You can also see ponds and windmills
Entrance fee :11,000 won (US$11)
Sungsan Sunrise Peak
A peak formed by volcanic eruption 100,000 years ago. One of the 7 wonders of nature
One of the best attractions in Jejudo
Entrance fee : 2,000 won (US$2)
Women Divers' Museum
A museum displaying women divers history, statues and art works.
Entrance fee : 1,100 won (US$1)
Udo (A small Island)
A small beautiful island located near Sungsan sun rise peak. You can rent a bike, ATV or moped to make a round this island. You need at least 3 hours for this island because of ferry schedule.
Ferry fee : 5,500 won (US$6 round trip)
Light house "All in" film location

You can see a church and a light house with open sea. The church was built for a television drama called "All in"
It has really beautiful landscape around.
No entrance fee
Aqua Planet
The biggest aquarium in South Korea
Recently opened. Ave. tour time is 2 hours
Entrance fee : 38,400 won (Anout US$38)
Trick Art Museum
The museum where you can take unique photos with 3D works of art. One of the very popular museums in jeju.
Entrance fee : 8,000 won (US$8)
Ilchul land
A large park where you can see beautiful garden, a cave and tropical plants. "Running men" film location and one of the popular destinations for Korean tourists
Entrance fee : 9,000 won (US$9)
Seungeup Folk Village
A traditional village preserved by government People are still living in this village and they are not allowed to renovate their houes.
No Entrance fee (Ave. Tour time :30 min)
Jeju Folk Village Museum

With a total of 117 houses and facilities, Jeju Folk Village Museum has restored the scenes of the village to that of the 1890's.
Also been a drama location for Daejangum
(Ave. Tour time : 1hour & 30 min)
Entrance fee : 9,000 won (US$9)

Cheonjeyon waterfall
You can see three waterfalls and a beautiful bridge where you can take a glimpse of Halla Mt. and the beautiful oean,
Entrance fee : 2,500 won (US$2.5)
Jungbang waterfall
A waterfall that falls straight down to the sea
Entrance fee : 2,000 won (US$2)
Cheonjiyon waterfall
A waterfall and a park. There is a large pond also with ducks.
Entrance fee : 2,000 won (US$2)

Seokwipo Submarine
You can ride a real submarine for an hour exploring the beautiful ocean of Jeju
Entrance fee : 56,500 won (US$56)
Seokwipo Pleasure Boat
This ferry take you to beautiful attractions like moon island, Jeongbang waterfall, Uwedolgye and bird island.
Entrance fee : 17,000 won (US$17)
Hueree Natural Park
Jeju nature park where you can see Dottong (traditional local bathroom), and millstones offer a glimpse of local life and culture.  For more info, please click here
Entrance fee : 9,000 won (US$9)
Uwedolgae Rock
A big rock with a legend. You can see beautiful sea and landscape.  I will tell you about the legend when you visit here. No entrance fee.
Very beautiful river-like sea. You can take a kayak or Jeju traditional boat called Teu.
No entrance fee but for Kayak, the fee is 10,000 won. Always long cue for this.
Secret garden shooting place

A shooting place for Korean drama called secret garden. You can see beautiful scenery around.
Entrance fee : Free
Jusaangjolli Cliff

A unique looking cliff formed by lava. Very poplular destination for tourists.
Entrance fee : 2,000 won  (US$2)
Yomiji Botanical Garden

Garden known as the best tourist attraction in Jeju Island. On the top of the garden, you can overlook the ocean and Seokwipo area
Entrance fee : 9,000 won (US$8)
Alive museum
This museum is alive. You will see all collections of 3D trick arts, digital arts, object arts, sculpture arts and provence arts.
Nice place for photo takings.
See video clip here
Entrance fee : 9,000 won (US$9)
Teddy bear museum
All collections of teddy bears.
Located at Seokwipo area. Most popular place
Entrance fee : 8,000 won (US$8)
Hello Kitty Island
Recently built. All about Hello Kitty.
Watch the video clip on youtube Click here
Entrance fee : 12,000 won (US$12)
Locadio World Studio
Recommendable if you are really into Korean dramas. A studio of Korean dramas and movies. You can take a photo feeling like you are an actor or an actress for Korean dramas. You can wear Korean traditional clothings also. Entrance fee : 9,000 won (US$9)
Health & Sex Museum - Seokwipo Area
The sexual museum where you can see sex related works of art and statues. Displays of
The cultures of sex in the world, sex education, sex fantasy and etc.
Entrance fee : 12,000 won (US$12)
Hallasan Summit (Halla Mountain)

The highest mountain in South Korea
It takes 8 hours to climb up and down.
No entrance fee. You can access there only Sungpanak and Goanhumsa routes.
Hallasan 4 hour Hiking Trail
Taking Erimpok entrance, you can hike in a Halla mountain for 4 hours. You cannot climb up to the summit from this entrance but you can be closer to it. Good for simple hiking in Mt. Halla.   No entrance fee.
Sara Orum
A crater located in the middle of Halla mountain. It takes 4 hours for round trip
No entrance fee.
Yakcheonsa Temple

A huge beautiful temple overlooking the ocean. You can see 80,000 of small buddhas
No entrance fee.
Hallim Park
A park where you can see many tropical plants, two small caves and folk village display. A very popular place for tourists
Entrance fee : 10,000 won (US$10)
Hyupjae Beach
A very nice and pristine beach where you can see a small island. It's a famous and popular beach in Jejudo during summer. Not recommended during winter.
No entrance fee.
Glass Castle
A museum that displays all kinds of glass arts
Very popular place to women tourists
Entrance fee :11,000 won (US$11)
Green Tea Museum
A museum for green tea & green tea plants. You can also buy fresh green tea & green tea ice cream, green tea bread and cosmetics
No entrance fee.
An unique looking mounstain near the ocean
Tthere is a buddha statue on the top.
You can also see a temple.
Entrance fee : 1,000 won (US$1)
Yongmeori Beach
- An unique looking cliff around the ocean
- You can make a round the beach area
Entrance fee : 2,000 won (US$2)
- A beautiful small mountaim with a great view
- A Korean drama, Daejanggeum Shooting place
Marado Tour
- A most southern island of Jeju isalnd
- You can make a round on foot.
- Average tour time 3 hours
Ferry fee : 15,000 won (US$15)

Jeju Nanta
Nanta performance in Jejudo

Entrance fee : VIP - 60.000 krw (US$60)
Regular seat - 50,000 Krw (US$50)
Starts at 17:00 or at 20:00
Mysterious Road
Your common sense isn't working here. Looks like a normal road but someting's different. You have to find out why.
No entrance fee
Rock of Dragon Head 
 A natural rock that looks like a  head of a dragon. Very popular site for many tourists

No entrance fee.
Natural History & Folklore Museum
A museum where you can see the history and ways of life of Jeju Island
Entrance fee : 1,100 won (US$1)
Dongmun Traditional Market
One of the big traditional markets in Jeju Island. Fish market, Fruit market, Souvenir market
No entrance fee
Starlight Universe Park
A park where you can learn about stars and constellations. There are 3D movie theater and huge telesopes. Good for kids 

Entrance fee : 5,000 won (US$5)