To reach the island by car as a starting point I take Trieste, which of course I recommend that you visit.
You must go toward the University of Engineering and from there continue towards the "boundary", cross the former customs Pese (Italy / Slovenia), Krvavi Potok and shortly after the bridge turn right onto Kozina, there you are on the E61.
I tell you now that Vignette or Slovenian MotorwayStamp you DO NOT need, don't look for alternative routes or sites that you want to show "the way" , it'just not needed.

At traffic light go straight toward Rijeka. After two hundred meters on the right side you'll found a distributor, go straight.

After about 45 minutes from Slovenia you should be near the customs Pasjak (Slovenia / Croatia), continue to Rijeka.You'll pass trough towns: Hrpelje, Materija, Markovščina, Gradišce, Obrov, Rusic, Podgrad, Racici, you just have to go straight.

After about 45 minutes from Slovenia you should be near the customs Pasjak (Slovenia / Croatia), continue to the direction for Rijeka. At sign Pula-Rijeka turn to right (Rupa)where starts the highway (the toll is paid in kuna - 5,7,10,20 kuna at the end of the way), in the case you traval by automobile are 5 Kunas.

Distance: 18 km Rijeka, Split 430 km, 104 km Pula.
Continue to Zagreb, Rijeka Split (Split, Zagreb, Rijeka).

After about ten minutes (respecting the speed limit) you will be at a super way and to the right you'll see large buildings and the sea. Go through the tunnels, on the fourth turn right, stay left, and you'll see the sign above that indicates Otok Krk, you're near then.

Now, you just have to do is follow the signs.

To reach island of Krk by bus I always consider Trieste as a starting point. Go to the bus station, is situated opposite of the train station ... I suppose you arrive by train ...

To get to Krk you must use the buss of "Autotrans", you'll take a bus at Trieste after you arrive to Rijeka, you have to change it for bus going to Krk. I carry you below a details of the departure times, arrival and prices in Kuna. I have taken it from the site of Autotrans.hr (not intuitive) on 22/07/2008 in the future it could change.

Departure from Trieste Arriving to Rijeka Line Price
13:30 15:54 Autotrans doo Autotrans doo 50.00
17:30 19:35 Autotrans doo Autotrans doo 50.00
18:30 20:49 Autotrans doo Autotrans doo 50.00
Leaving from Rijeka Arriving to Krk Line Price
06:15 07:37 Autotrans doo Autotrans doo 52.00
08:15 09:37 Autotrans doo Autotrans doo 52.00
09:30 10:52 Autotrans doo Autotrans doo 52.00
11:00 11:58 Autotrans doo Autotrans doo 52.00
12:00 13:22 Autotrans doo Autotrans doo 52.00
13:45 15:07 Autotrans doo Autotrans doo 52.00
14:30 16:26 Autotrans doo Autotrans doo 52.00
14:30 15:28 Autotrans doo Autotrans doo 52.00
15:15 16:37 Autotrans doo Autotrans doo 52.00
16:15 17:37 Autotrans doo Autotrans doo 52.00
17:30 18:52 Autotrans doo Autotrans doo 52.00
18:15 19:30 Autotrans doo Autotrans doo 52.00
19:30 20:52 Autotrans doo Autotrans doo 52.00
21:15 22:35 Autotrans doo Autotrans doo 52.00

Take the ferry to arrive to Krk

You can do it but it's a little more complicated and in my humble opinion is not worth it. Indeed you must take two or three ferries in addition to travel by car to reach the road in the various ports. In recent years Croatia has made great strides in building roads and motorways and have no intention of stopping. However I'll give you a few examples:

1.from Venice you can reach Rabac, from Rabac by car you must go to Brestova take the ferry for Porosina (Island of Cres), then you have to go by car to Mareg (Island Cres) and from there take the ferry for Valbiska (Island of Krk );
2.always from Venice you can to arrive to Mali Lošinj, in the car driving up to Mareg (Island Cres) and from there take the ferry for Valbiska (Island of Krk);
3.other alternatives that involve a departure from Pescara, Bari or Brindisi ...in my opinion are to be excluded unless you intend to visit all the Croatian coast from south to north;