To get to Brijuni (well, just to Veli Brijun, the biggest of the islands in Brijuni archipelago) and enjoy the island on your own you will need to catch the boat from a small fishing town Fazana, few kilometers North of Pula.
Here's the Brijuni NP office where you will buy the tickets - on the main street, but also your hostel in Pula may help you if you ask - in high season this may be good thing to consider.
So far the April wasn't the crowdest of months and we didn't reserve the tickets (both park ticket and ferry ticket are one) in advance; we went straight to the office and bought them for next day, but if you go to NPs web site they say you have to make reservation before you'd intend to go. Actually, we were staying in a camp near and hadn't have that issue, but if you come from a far and in high season, make sure you visit their web site and collect the information, plus reserve the visit as the boat is quite small - there's also information about departures and prices.
Note that there are touts around Fazana (and Pula, Rovinj, etc) fishing for potential customers - that's for different kind of sightseeing, the panoramic one - without actually stopping on the ground of the Big island.
Well I think the best way to explore the island is by bike, because if you walk some things are little far, but it's doable if you have plenty of time (let's say 1 full day, depends how fast you walk, how much attentnion you invest into sights of your interest, what you want to see).
Maybe the 'worst' of the transport options is tourist 'train' (tractor with trailer) for which most people opt; you're part of a large group and there aren't best conditions to take photos if you sit in a wrong side. For snaps it's just fine, as well as if you don't like to walk or cycle. I also noted they don't stop in some of the interesting places or it's just too fast leaving some really nice spots. They don't charge extra for it as you already paid a NP ticket (I was little disappointed that price for a visitor to NP we bought from an office of the park in Fazana has no 'discount' for those who wish to visit without being part of a guided tour, for all (foreign?) adults at that time (low season in April 2011) was 170 KN/person or almost 23 Euro).
Unfortunatelly, not your own one - these are not allowed to transport on a passenger boat from Fazana. You can rent it from a sport office next to hotel Neptun - the one classy building that sees Brijuni port. It's 25 KN/person per hour.
Last option would be a mini car to rent, that kind of car that would be used for golf grounds - if you know what I mean.
Altogether there are 274 kilometers of roads and unpaved or paved paths - and among that only 24 kilometers are paved.
Ako se odluciš iznajmiti ovakav autic, onda uz isti ide i vozac. Ja to nisam probao jer sam, neznajuci obicaje, otokom projurio u turistickom vlakicu što se pokazalo kao pogreška.
Za razliku od brijunskog expressa, koji pomahnitalo juri otokom kao da ga ganjaju gladni lavovi, koristeci ovaj autic ti sam odreðuješ ritam vožnje i mjesta na kojima ceš se zaustaviti. Po mojem sudu to je kud i kamo povoljnija opcija za razgledavanje.
Definitivno, bike je najbolje sredstvo za obilazak otoka. Ako k tome iznajmiš tandem, posao ti je upola lakši. Ovo društvo Talijana, njih tri para na tandemu, mora da su uživali u obilazku.
Na moju žalost, prepustio sam se raljama vodica umjesto da se raspitam što se nudi i kakve su opcije.
Tebi koji ovo citaš, a imaš namjeru posjetiti Brijune, toplo preporucam da ne pociniš moju grešku. Iznajmi bike i uživaj na svoj racun.