There are now different kinds of cards split similar to these, and some of the rules have changed since the original question was asked.
Basic Split Cards
For example: Fire // Ice. These cards are read with an “and” joining the two names, thus “Fire and Ice”.
These cards work exactly the way you asked, you cast one half of the card for the casting cost, get that effect and the whole card goes to the graveyard. You don't have the option to cast the other half.
Fuse Split Cards
Example: Turn // Burn. These follow the same naming pattern as basic split cards, thus “Turn and Burn”.
In Return to Ravnica block, a new ability was given to split cards, fuse, which allowed you to cast both halves. The cost would be the combined cost of both halves, they would be put on the stack as a single spell and would resolve starting with the left half (which usually made the right half work better anyways). You could chose to play only one half of the split spell.
Aftermath Split Cards
For example: Never // Return. These cards are read with a “to” joining the two names, thus “Never to Return”.
In Amonkhet block, cards with another new ability, Aftermath were printed. These cards had the top half printed as a spell normally, with the bottom half turned sideways. You could only cast the top half normally, the bottom half would be cast like flashback, from the graveyard then exiled.
Common Split Card Rules
The rules for split cards changed, the most recent change was when Aftermath cards were created in Amonkhet. Before the cards would be seen as both separate and combined by effects letting Wear // Tear be seen as a 1, 2 and 3 CMC spell. This let some spells be abused by using the smaller half to meet the imprint requirements of Isochron Scepter but casting the higher half, or using Counterbalance to get three CMCs worth of counter power. Now the CMC of split cards is always the combined CMC of both halves, EXCEPT while the spell is being cast and on the stack, where only the parts paid for are counted. The same is true for the colors of the spell, the colors of a split card are both colors everywhere EXCEPT on the stack when being cast, where only the cast part(s) colors are used.
This is covered in the comprehensive rules:
708.4 In every zone except the stack, the characteristics of a split card are those of its two halves combined. This is a change from previous rules.
708.4a Each split card has two names. If an effect instructs a player to choose a card name and the player wants to choose a split card’s name, the player must choose one of those names and not both. An object has the chosen name if one of its names is the chosen name.
708.4b The mana cost of a split card is the combined mana costs of its two halves. A split card’s colors and converted mana cost are determined from its combined mana cost.
Example: Assault/Battery’s mana cost is {3}{R}{G}. It’s a red and green card with a converted mana cost of 5. If you cast Assault, the resulting spell is a red spell with a converted mana cost of 1.
708.4c A split card has each card type specified on either of its halves and each ability in the text box of each half.
Adventure Cards are not Split Cards
For example: Brazen Borrower
Adventure cards are not split cards, but are still two spells on one card. They have a main spell that is a creature, and an adventure spell that is an instant or sorcery. The text box shows the creature side on the right, and the adventure spell on the left.
You can cast either half as normal. If you cast the adventure spell, the card is exiled as it resolves and the creature half can be cast from exile. If you cast the creature half from hand, it resolves and you cannot cast the adventure spell.
These do not follow the split card rules listed above, since they are not split cards; instead they follow the Adventurer rules. Adventure cards are treated as if they are only the creature half at all times except when they are on the stack being cast as the adventure.