The Marutto Fukuoka is basically two travel passes in one: you get a one-day train pass and a one-day bus pass. While you could use them on the same day, you don't have to, meaning this 2000 yen pass can cover at least most of your travel expenses over two days.
The pass can be purchased at the local airport (international terminal), or at the tourist information at Tenjin Station (by the expressway bus terminal), or at Nishitetsu Fukuoka (Tenjin) Station. I bought it at the tourist information by the bus terminal, and I had to pay cash; I'm not sure if the airport would accept credit cards, so be sure to have cash on hand when buying. Passports have to be presented to purchase since it's for visitors only.
The bus pass allows you to ride buses within Fukuoka city for a day. It's a scratch card, so you just have to scratch off the year, month, and day you want to use it, and show it to the bus driver when getting off the bus.
In addition to the passes, you'll get a booklet that features sample itineraries for the train and bus passes, as well as coupons for select stores and restaurants.
Yanagawa is a canal town about an hour from Tenjin: if you catch a limited stop train, it'll be a bit quicker (50 minutes). Like other canal towns around Japan (e.g. Otaru, Kurashiki) it's a small but pretty town with a handful of historical buildings and the opportunity to ride a gondola down the canal. It's worth noting that Yanagawa Station is fairly far from the city centre: if you want to take the boat ride, you can buy tickets at the bus stop right next to the station, and a shuttle bus will take you to the boat drop-off point. Otherwise, you'll have to take a local bus (or walk), and as in most small towns, buses here are infrequent.
That said, Yanagawa is a pretty town to walk around slowly to enjoy the atmosphere, especially if you're looking for a break from the bustle of the big city of Fukuoka. I would advise checking if any spots you want to visit are closed on certain days: we unfortunately had to skip the former Toshima residence since we came on a Tuesday and it turned out to be closed on Tuesdays. I also want to add that for anyone interested in the Tachibana Garden, that you cannot actually go into the garden: the entrance fee gets you access into the house, and then you can view the garden from a balcony.
Incidentally, the local specialty appears to be unagi, which are about as expensive as you'd find anywhere else. If you're on a budget, you can buy unagi onigiri as a snack.
Once again, bus service is infrequent, so if you're on a tight schedule you may want to check bus schedules in advance. Otherwise, it's about a 30 minute walk from the city centre to the train station.
Afterwards I went to Dazaifu, which required a transfer at Futsukaichi Station. Dazaifu is a popular enough tourist spot that when approaching Futsukaichi, there will be an announcement in English as a reminder to transfer here if you're heading to Dazaifu, and once you've gotten off the train there will be signs in English directing you to the right platform for Dazaifu.
Unlike Yanagawa, Dazaifu's main attractions are all very close to the station: there are plenty of signs in English pointing which way to go to reach Tenmangu Shrine. Incidentally, Tenmangu Shrine is known for enshrining the god of scholarship, so if you or anyone you know could use some praying for school... this might be a good stop to make, to pray and maybe buy a good luck charm. You'll likely see big groups of bus tours as well as students who've come to pray for their studies. It's definitely much busier than Yanagawa is. The walk from the station to the shrine is a shopping arcade that sells plenty of souvenirs, as well as some sweet shops: the local sweet is umegaemochi, or grilled mochi filled with red bean.
The following day, I used the bus pass to go around Fukuoka, hopping on and off at various spots that looked interesting. There are lots of things to see in Fukuoka, and the bus pass covers a wide area: it's a good opportunity to go from one end of the city to the other (or at least far from wherever you're staying in the city) without having to pay bus fare, since bus fares here are calculated by distance. If you enjoy gardens, I would highly recommend the Japanese garden in Ohori Park, which incorporates elements from Japanese and Western gardens. The adult entrance fare is 230 yen, but I believe there is a special discount for foreigners: the woman at the ticket counter asked where we had come from, and when we told her, she informed us that it was only 190 yen. I'm not sure if this is a temporary or permanent fare, but I think even 230 yen would be a fair price for such a well-tended garden. (On another note: the garden does have bathrooms, but women should note that the women's toilet is a squat toilet, not a western toilet. They do have a handicap washroom that has a western toilet, though.)