Social Share Links with Custom Icons
Your required social media custom share links are provided below.
http://www.facebook.com/sharer.php?u=[EncodedURL]
http://twitter.com/share?text=[TITLE]&url=[URL]
Google +
https://plus.google.com/share?url=[EncodedURL]
http://www.linkedin.com/shareArticle?mini=true&url=[EncodedURL]
http://pinterest.com/pin/create/button/?url=[EncodedURL]&media={[MEDIA]}&description=[TITLE]
You can find some other social media custom URL links and implementation guide from here - How to Create Custom Social Share Links
The solution suggested by Grzegorz is deprecated and it's not really the best way to do it. For Facebook, you just need to load the JS SDK ( https://developers.facebook.com/docs/reference/javascript/ ) and call the UI feed method on click on your custom button. More details on the UI method: https://developers.facebook.com/docs/reference/javascript/FB.ui/
For Twitter: https://dev.twitter.com/docs/tweet-button#build-your-own
You will have other options for LinkedIn, Google+, etc. but I don't know them from the top of my head and not sure if of all them offer alternatives.
Keep searching, you probably just didn't dig deep enough, it took me 2 minutes to find the link for Twitter :)
Social link URL’s
https://twitter.com/intent/tweet?text=[TEXT]
http://www.facebook.com/share.php?u=[URL]&title=[TITLE]
StumbleUpon
http://www.stumbleupon.com/submit?url=[URL]&title=[TITLE]
Delicious
http://del.icio.us/post?url=[URL]&title=[TITLE]]¬es=[DESCRIPTION]
http://www.linkedin.com/shareArticle?mini=true&url=[URL]&title=[TITLE]&source=[SOURCE/DOMAIN]
Technorati
http://technorati.com/faves?add=[URL]&title=[TITLE]
Tumblr
http://www.tumblr.com/share?v=3&u=[URL]&t=[TITLE]
http://www.reddit.com/submit?url=[URL]&title=[TITLE]
Google Bookmarks
http://www.google.com/bookmarks/mark?op=edit&bkmk=[URL]&title=[title]&annotation=[DESCRIPTION]
Evernote
http://www.evernote.com/clip.action?url=[URL]&title=[TITLE]
Just a quick update on these links, here's a link for Google+
https://plus.google.com/share?url=[URL]
Now, if you want those links to open in a new window, just add this javascript code after the href's:
onclick="javascript:window.open(this.href,'', 'menubar=no,toolbar=no,resizable=yes,scrollbars=yes,height=600,width=600');return false;"
This javascript code works with Twitter, Google+ and Facebook (maybe some other, but I haven't tested any other social networks).
Example for WordPress:
<a href="https://plus.google.com/share?url=<?php the_permalink(); ?>" onclick="javascript:window.open(this.href,'', 'menubar=no,toolbar=no,resizable=yes,scrollbars=yes,height=600,width=600');return false;">Here you can add text, image, whatever.</a>