You should leave one side empty, hence the name "partial range".

let newStr = str[..<index]

The same stands for partial range from operators, just leave the other side empty:

let newStr = str[index...]

Keep in mind that these range operators return a Substring. If you want to convert it to a string, use String's initialization function:

let newStr = String(str[..<index])

You can read more about the new substrings here.

You can get the ranges of your substrings using a while loop to repeat the search from that point to the end of your string and use map to get the substrings from the resulting ranges:

extension StringProtocol {
    func ranges<S:StringProtocol,T:StringProtocol>(between start: S, and end: T, options: String.CompareOptions = []) -> [Range<Index>] {
        var ranges: [Range<Index>] = []
        var startIndex = self.startIndex
        while startIndex < endIndex,
            let lower = self[startIndex...].range(of: start, options: options)?.upperBound,
            let range = self[lower...].range(of: end, options: options) {
            let upper = range.lowerBound
            ranges.append(lower..<upper)
            startIndex = range.upperBound
        }
        return ranges
    }
    func substrings<S:StringProtocol,T:StringProtocol>(between start: S, and end: T, options: String.CompareOptions = []) -> [SubSequence] {
        ranges(between: start, and: end, options: options).map{self[$0]}
    }
}

Playground testing:

let string = """
your text
id:244476end
id:383848448end
id:55678900end
the end
"""

let substrings = string.substrings(between: "id:", and: "end")  // ["244476", "383848448", "55678900"]

You can split the string and get the first part like this:

var str = "Tower Capital, 99822, Building 2399"
let firstItem = str.componentsSeparatedByString(",").first

Or when using Swift >= 3:

str.components(separatedBy: ",").first

The problem is that mentions[0...3] returns an ArraySlice<String>, not an Array<String>. Therefore you could first use the Array(_:) initialiser in order to convert the slice into an array:

let first3Elements : [String] // An Array of up to the first 3 elements.
if mentions.count >= 3 {
    first3Elements = Array(mentions[0 ..< 3])
} else {
    first3Elements = mentions
}

Or if you want to use an ArraySlice (they are useful for intermediate computations, as they present a 'view' onto the original array, but are not designed for long term storage), you could subscript mentions with the full range of indices in your else:

let slice : ArraySlice<String> // An ArraySlice of up to the first 3 elements
if mentions.count >= 3 {
    slice = mentions[0 ..< 3]
} else {
    slice = mentions[mentions.indices] // in Swift 4: slice = mentions[...]
}

Although the simplest solution by far would be just to use the prefix(_:) method, which will return an ArraySlice of the first n elements, or a slice of the entire array if n exceeds the array count:

let slice = mentions.prefix(3) // ArraySlice of up to the first 3 elements