Shibuya and Shinjuku are just 6 minutes of Yamanote away (and it's the slowest line). From Shinjuku, almost every train southbound stops at Shibuya. There's one Yamanote train every 3 minutes, and with all the other lines you really don't have to worry about train departure times and stops, there's a constant flow of trains and people from Shibuya to Shinjuku. Really, it sounds silly, but you'll just have to walk to the station and take the first train! It's easier than it may look.
But if you're particularly paranoid, check this out and see for yourself.
Enter the station name
About tickets. One trip from Shinjuku to Shibuya costs 150yen. You can buy single tickets or get a pre-paid card. There are 2 cards, PASMO and SUICA. The second one only works on JR trains, PASMO works almost everywhere. They cost just 500yen, you can buy credit with ease from any automatic ticket venging machine. You'll pay always the full price if you use them without a monthly ticket. SUICA doesn't have a monthly pass, only PASMO has. When you buy a rail pass, you pay for a certain trip - for example, you can get a monthly pass from Shibuya to Shinjuku, but you'll pay every trip outside these two stations at full price.
More stuff:
Suica - Wikipedia, the free encyclopedia
Tokyo Prepaid Cards: Suica, PASMO
PASMO
PASMO - Wikipedia, the free encyclopedia
Don't bother using Tokyo metro except for those places which aren't covered by JR trains.
Don't worry about a thing, trains are very easy to use. Get a Pasmo or a Suica and use it without worrying about monthly passes, they'll end up in costing you more - they're worth only if you plan to move *only* from a certain station to another, if you want to move around the city buy a card, put 30.000yen in there and just go wherever you want. You'll have to worry about your credit only after two months, maybe less, maybe more...