Best time to visit is May - September, after the rains, and before the lakes dry out. The almost constant breeze may fool you into thinking the sun is not getting at you so be super careful and remember to constantly apply sun screen!
from São Luís(easiest), take a bus to Barreirinhas (which has the best structure). There are a number of different tour companies that can arrange to have you collected from your accommodation in Sao Luis (hotel, hostel or even your own house!) and will take you by luxury coach to Barreirinhas where you will then transfer to a 4X4 to continue the journey to the national park. Prices are reasonable enough at around R$150 for the day. You will leave Sao Luis at about 4am and be back at 9pm. The day trip is long, but absolutely worth it if you don't want to stay overnight closer to the national park. Having Portuguese would be a rather beneficial tool as not many of the drivers, tourism employees, travel companies spoke English.
You can also just get a regular bus from the Sao Luis rodoviaria to Barreirinhas, which is not expensive. The trips into the Lençois from Barreirinhas are very touristy. For a more authentic and personal experience, get the boat from Barreirinhas to Atins, and from there a 4x4 (try Pousada de Buna) and a guide to take you to the lagoon.
Get a 4x4 directly to Barreirinhas which takes around 7 hours, costs R$1200 for up to 4 people. The Brasil Discovery agency in Barreirinhas  may have a solution much cheaper if they have someone in Jeri the day you leave. There are also several places that offer a tour of 3 days for the Lençois for R$1500 (only transport). Bargain the price, especially during low season!
Use Public transport. Get buses and vans from Jeri to Barreirinhas or Cabure via Camocim, Parnaia, Tutoia, Paulino Neves. It takes two days (one night in Parnaiba or Tutoia) and costs around R$150. You can book part of this trip at Global Connection in Jeri. In addition, Global Connection offers 4x4 transport with cars for 4 or 6 people, directly to Cabure, from where you can cross the river to Atins. Prices between R$900 and R$1200 depending on car (4 or 6 people) and conditions. They will also try to combine clients in cars, so that this trip might be more affordable. You can find Global Connection on Rua do Forro in Jericoacoara. They will tell you all possible options. Contact: email@example.com / (88) 99900 2109
Most people sleep in Barreirinhas and do daytrips. However, the dunes are far more accessible if you stay in Atins, or even better in Canto de Atins. Mandacaru and Caburé are less convenient alternatives. Paulino Neves offers proximity to similar sights (Pequenos Lençóis), although not inside the actual national park. There is also the little explored town of Santo Amaro on the park´s western rim.