A fast lens (large maximum aperture) will always be useful - but is not essential.
A small lens can be useful - but is not essential. I usually use a less than tiny 17-250mm omnFor an APSC camera. It has seldom been of vast disadvantage. I also sometimes usevery strongly recommend a 50mm. f /1.8 prime - about as smallzoom with a lensminimum focal length of around 17 or 18mm and a top end of as theremuch as you can afford at the quality level desired. ie 17-55mm is an excellent start, but 17-100+mm will not go astray, and I find it of no great benefit size wisean 18-250mm very useful. Details, reasons and examples below ...
A prime lens may be "classic" - but reduces your optionsThere are a number of traditional recommendations. IfThey have their place, but if you want to be 'arty' or to pursue some subset ofbest possible photoscomposition results in all situations, then a prime may meet your needs. If you want flexibility18mm to deal with anything you may wishxxx is preferable to photo graph, then as wide a zoomless range as possible is desirableor higher minimum focal length + a classic solution.
A fast lens (large maximum aperture) will always be useful - but is not essential.
A small lens can be useful - but is not essential. I usually use a less than tiny 17-250mm on an APSC camera. It has seldom been of vast disadvantage. I also sometimes use a 50mm. f /1.8 prime - about as small a lens as there is - and I find it of no great benefit size wise.
A prime lens may be "classic" - but reduces your options. If you want to be 'arty' or to pursue some subset of possible photos, then a prime may meet your needs. If you want flexibility to deal with anything you may wish to photograph, then as wide a zoom range as possible is desirable.
Street photography usually does not emphasise utter lens performance or adherence to rules - contents and inter-relationships usually predominate over absolute crispness of focus, spot on exposure of rule-matching framing. So while having the best that Carl-Zeiss can make would be very nice, a kit lens may not be too bad a start.
Starting with your 17-55mm kit lens, equivalent to a full frame 26 - 83 mm will probably be a good startinitial choice.
I mostly use an 18-250mm f/3.5-6.3 on an APSC body = full frame 27mm- 375mm. The topntop end gets used less often than the 18mm, and most photos would be in the 18-100 (27-150 equivalent) range. Very few people will recommend a 17 or 18mm lens and it is often cited for producing purposeful distortion, but I find it superb for super closeup 'fly on the wall' photos and for getting a good width of scene in restricted situations and when you need to act rapidly. With experience you can 'shoot from the hip' while walking through a crowd of people, or raise the camera slightly, and stand right in amongst a group of people - as invisible or visible as you wish to be. Sure - people see a tourist with a biggish camera, but you are not obviously an "in the face" photographer. Too too close and you get perspective distortion. Back off a small way and it becomes acceptable.
The following examples will notThe examples in this album were all betaken at 18mm focal length using an APSC camera - in most cases a Sony A77 or A700. In many but this isnot all cases I have added rectangles which show what you would see if a 28mm lens had been used from the sort of shotsame camera position. In the examples where I have not included a 28mm crop rectangle you could typically getwill have a good idea of the effect after having seen numerous other pictures with the 17-55 kit lensrectangles included.
Larger versionHere is an example set of 4 photos, all taken at 18mm and with 28mm crop rectangles shown. - Windows: right click(18mm and select open in new tab28mm on an APSC corresponds to about 27mm and 42mm on a full frame 35mm sensor.) A a prime 35mm lens would cover less area again than the 28mm and a 50mm even less again.
More anon maybeIn the alleyway shot, top right, from the position used, a 28mm lens would have vastly reduced the framing options. With even a 17-55mm zoom you could choose from the framing shown down to an area about half the side length (1/4 the area) of the red rectangles.
In the photo of the man on the tricycle, from the position used, the difference between 18mm and 28mm is vast. The 28mm cropping is too tight for even a shot focusing exclusively on the tricycle.
Similarly, the people shots at top left and bottom right offer vastly improved flexibility with a non prime lens with a 17mm minimum focal length.
In some cases "foot zoom" will allow you to increase image coverage but this is often not possible when spontaneity or rapid action are required. In the group portrait at top left, you'd need much longer arms :-).
See also my different but relevant answer to Any reason to buy stock lens for Sony Alpha if I already have good Minolta Maxxum lenses?