You are not the only one who couldn't find the solution.
String doesn't implement RandomAccessIndexType. Probably because they enable characters with different byte lengths. That's why we have to use string.characters.count (count or countElements in Swift 1.x) to get the number of characters. That also applies to positions. The _position is probably an index into the raw array of bytes and they don't want to expose that. The String.Index is meant to protect us from accessing bytes in the middle of characters.
That means that any index you get must be created from String.startIndex or String.endIndex (String.Index implements BidirectionalIndexType). Any other indices can be created using successor or predecessor methods.
Now to help us with indices, there is a set of methods (functions in Swift 1.x):
Swift 4.x
let text = "abc"
let index2 = text.index(text.startIndex, offsetBy: 2) //will call succ 2 times
let lastChar: Character = text[index2] //now we can index!
let characterIndex2 = text.index(text.startIndex, offsetBy: 2)
let lastChar2 = text[characterIndex2] //will do the same as above
let range: Range<String.Index> = text.range(of: "b")!
let index: Int = text.distance(from: text.startIndex, to: range.lowerBound)
Swift 3.0
let text = "abc"
let index2 = text.index(text.startIndex, offsetBy: 2) //will call succ 2 times
let lastChar: Character = text[index2] //now we can index!
let characterIndex2 = text.characters.index(text.characters.startIndex, offsetBy: 2)
let lastChar2 = text.characters[characterIndex2] //will do the same as above
let range: Range<String.Index> = text.range(of: "b")!
let index: Int = text.distance(from: text.startIndex, to: range.lowerBound)
Swift 2.x
let text = "abc"
let index2 = text.startIndex.advancedBy(2) //will call succ 2 times
let lastChar: Character = text[index2] //now we can index!
let lastChar2 = text.characters[index2] //will do the same as above
let range: Range<String.Index> = text.rangeOfString("b")!
let index: Int = text.startIndex.distanceTo(range.startIndex) //will call successor/predecessor several times until the indices match
Swift 1.x
let text = "abc"
let index2 = advance(text.startIndex, 2) //will call succ 2 times
let lastChar: Character = text[index2] //now we can index!
let range = text.rangeOfString("b")
let index: Int = distance(text.startIndex, range.startIndex) //will call succ/pred several times
Working with String.Index is cumbersome but using a wrapper to index by integers (see https://stackoverflow.com/a/25152652/669586) is dangerous because it hides the inefficiency of real indexing.
Note that Swift indexing implementation has the problem that indices/ranges created for one string cannot be reliably used for a different string, for example:
Swift 2.x
let text: String = "abc"
let text2: String = ""
let range = text.rangeOfString("b")!
//can randomly return a bad substring or throw an exception
let substring: String = text2[range]
//the correct solution
let intIndex: Int = text.startIndex.distanceTo(range.startIndex)
let startIndex2 = text2.startIndex.advancedBy(intIndex)
let range2 = startIndex2...startIndex2
let substring: String = text2[range2]
Swift 1.x
let text: String = "abc"
let text2: String = ""
let range = text.rangeOfString("b")
//can randomly return nil or a bad substring
let substring: String = text2[range]
//the correct solution
let intIndex: Int = distance(text.startIndex, range.startIndex)
let startIndex2 = advance(text2.startIndex, intIndex)
let range2 = startIndex2...startIndex2
let substring: String = text2[range2]
