I'm not a fan of the current so-called "simplified" permissions in the Play store, and would like a way to check the exact permissions of an app before installing. This need not be a tool that does the installation itself, I just need a site or app that helps me look up the exact permissions from the app before I install it. Is there such a tool available?
You don't need an external tool for this. On the versions of the Play Store with the new "simplified" permissions dialog, you can still find a full list of permissions from the app's store page (the screen with the icon at the top, description, screenshots, &c.). Scroll down to the bottom of the page. Under "Additional Information" you'll see "Permissions" with a link "View details". Click this link to see a dialog with the list of individual permissions.
However, if you do want an external tool, AppBrain is an alternative web front-end to the Play Store, and lists the permissions individually for each app. From there you can click through to install the app from the real Play Store. Not all apps appear on AppBrain, but many of them do, especially the popular apps.
There's also a resource offered by Izzy, a good friend of this site. His website offers a curated list of apps, searchable by permissions. It also gives a fuller description of each permission, and highlights interesting combinations. Because apps are added to his site by a manual process, it's a very small list compared to the huge number of apps on the Play Store, but consists of a good range of recommended and popular apps in different categories.
In his answer, Dan already pointed to my list of apps by real-life topics, which offers some extras as well. Currently, it only covers a small subset of what's available on Google Play, though (for some numbers: ~10,000 apps = ~1% of Google Play, including roughly 5% of the apps available in Aptoides (curated) main repository, and a third of the apps available on F-Droid – with some overlappings).
One more resource to mention for apps not covered there is Appbrain, which is a different front-end to Google Play Store. So you should find all apps there, and they also show "concerns" (what Dan put as "interesting combinations") – a very useful thing to be warned about potential danger. If an app is missing from their list, you can easily make it appear by copy-pasting its package-name (or full Google Play URL) on this page. It might take some minutes then, but it will eventually be available (usually within 10..30 minutes).
Galen recently also introduced his app "StripSearch". I've used it a little (and had an intensive discussion with him – partly playing "beta tester" ;) This app not only shows you the permissions before installing an app – it even allows you to filter your app searches by permissions, e.g. to exclude unwanted ones (privacy concerns) or specifically require others (such as using the real flash-light when in need of a flashlight app, practically excluding those using the display or other means). Runs pretty well, definitely worth a try. You can find StripSearch at Google Play.
While this isn't used for viewing permissions (better off following the advice given by the other answers), I feel that I am obligated to endorse XPrivacy , an Xposed module for restricting permissions. If you are ever worried about extensive permissions, use this. Its completely open source too!
I think that a mixture of XPrivacy and PlayPermissionsExposed will do the trick. They are part of the popular Xposed framework. For GB devices, you can use liudongmiao's Xposed Installer GB at the XDA developers forum.
Make SURE to read the instructions. I'm not responsible if you do not.
If you run into any problems, please feel free to comment.
PS: The XPosed framework needs root, unless you manually decide to flash in recovery. If you are not willing to do so, look at the other answers.