There are several options to get to Beppu.
By Plane
Via Oita airport: Oita airport is the closest airport to Beppu.
Currently 14 daily flights are operated from/to Tokyo Haneda. Also, some regular flights are also operated from/to Tokyo Narita, Osaka Itami, Nagoya Chubu, and Seoul Incheon (Korea).
Via Fukuoka airport: Fukuoka is the major city in the Kyushu island, from which Beppu can be reached in about 2 hours by bus or train. A great number of daily flights are operated between Tokyo Haneda and Fukuoka. Also, daily flights are operated from/to other major airports in Japan such as Tokyo Narita, Osaka Itami, Osaka Kansai, and Nagoya Chubu. There are also international direct flights from/to Helsinki (starting in May 2016), Seoul Incheon, Beijin, Shanghai, Hong Kong, Taipei, etc.
By Train
From Tokyo, Osaka, and Nagoya: After arriving at one of these cities by Plane, take Shinkansen bullet train to the direction of Hakata (westbound), and get off at the Kokura Station, which is the first station Shinkansen bullet trains make a stop in the Kyushu Island. Then, change to the JR Nippo Main Line and take Sonic limited express train to the direction of Oita (eastbound), and get off at the Beppu station.
By Ferry
An overnight ferry is daily operated between Osaka and Beppu.
From Fukuoka Airport:
Train
Take subway to Hakata Station (subway).
Go up to the ground level to Hakata Station (JR), and from there, take a train that goes to Beppu station. The price/time depends on which train you use. We recommend to use the Sonic limited express train which takes approximately 2 hours (time table here). The regular cost is 5,050 JPY one-way including base fare ticket and express ticket (no seat reservation). A discounted ticket called "Ni-mai-kippu" (2 tickets) can be bought for 6,180 JPY and can be used either way (2 people can use it for one-way, or one person can use both for return), and includes seat reservation fees. (There is also a "Yon-mai-kippu" (4 tickets) for 11,000 JPY with the same conditions). These tickets can be bought via vending machines, or at the ticketing office.
Bus
There is a bus service from the International Terminal of Fukuoka Airport to Beppu Kitahama. It is a short (5+ min) walk from Beppu Kitahama to Beppu Station. Time table is here. The price is 3,190 JPY one-way, and 5,600 JPY for return.
From Oita Airport:
Bus
There is a bus service from the airport. Take a bus bound for Beppu Kitahama (or Beppu station). The ride is approximately 50 minutes. Tickets can be bought at vending machines (cash only). The prices is 1,500 JPY one-way, 2,600 JPY for return, and 4,100 JPY for coupon tickets (4 tickets). Time table from the airport is available here for Oct 16th and here for other dates (note that the departure time from the airport is adjusted to arriving flights, so the times may change if there are delayed flights). It is a short (5+ min) walk from Beppu Kitahama to Beppu Station. (Time table to the airport is here).
Taxi
Taxi service from Oita airport to Beppu costs approximately 10,000 JPY, and 12,000 to Suginoi Hotel. The exact cost will be measured by the meter. The rate is increased for late night. Note that most taxis will only accept cash.
From Beppu Station:
There is a (small but) free shuttle bus that runs to/from Beppu Station and Suginoi Hotel.
Practical Information
Public phones are rare. See here for information on mobile services (SIMs/WiFi access) in Japan.
Tipping is never expected.
The voltage of electricity is 100V. Plug & sockets are type A (2 pins - the same as in North America).
Tap water is drinkable.
Credit cards can be used somewhat, but many shops still only accept cash. Cash can be withdrawn with credit card from ATMs of Seven Bank (located in practically all Seven Eleven stores), or Japan Post. Other ATMs may only accept credit cards that have been issued in Japan. One of the ATMs nearest to the venue is in the Seven Eleven store just across the street from the entrance of the Hotel (Hana-kan). Another is here.
You can search for flight/train connections at the following site:
Train Route Finder by Jorudan Co,. Ltd.