It's a bit unclear to me as well how you actually do your brewing, but I'll try to describe as accurately as possible how I do it. I've made about a dosen batches so far without any problems (some with just malt extract, some with malt extract and sugar) .
If you are even slightly in doubt of the water quality in your area, boil and cool all the water you use. Alternatively you can use bottled water.
Clean all the items you are going to use. After cleaning sanitize them with a sanitizing agent (preferrably something especially made for home brewing). Bleach should work as well, but I've never used it so I can't advise you on how to use it.
Remember to clean and sanitize the fermenting buckets as well. Its best to have two buckets, but you can make beer with just one.
Mix the yeast to a small amount (about 2 desilitres) of warm water (35 degrees celsius), cover it with plastic wrap and leave it alone for 15 minutes.
Boil a small amount (half a desiliter) of water and dissolve a teaspoonful of sugar or malt extract into it. Cover and leave to cool to 35 C. After the yeast has been in the water for 15 min and the sugar water has cooled, pour the sugar water into the yeast water.
In half an hour (probably sooner) there sould be some foaming/churning to be seen in the yeast and sugar water, this means the yeast is alive and well. If nothing is happening, the yeast is probably dead. If so, do all again from the beginning with a new yeast.
The above described yeast proofing method is not strictly required, but will give best results since the yeast has an opportunity to wake up before pitching into the wort. Also you will know for sure if the yeast is alive or not.
Put the extract (and sugar if used) into the sanitized fermenter. Pour the advised amount of boiling water onto them and dissolve completely.
Add as much cold water as the intructions tell you to.
Pour the wort to another fermenting bucket and back again a couple of times. This will aerate the wort so that the yeast has the oxygen it needs in the beginning of the fermentation. If you don't have two fermenting buckets, close the lid of the bucket and shake it vigorously for a while.
Check that the temperature of the wort is in recommended range and pitch in the yeast water. Seal the fermenter and pour some water (or vodka) into the airlock.
Leave the fermenter alone for two weeks instead of the one week advised in most beer kit instructions. This way the conditioning time in bottles doesn't have to be as long, and the beer will taste better after a shorter time than usually.
After two weeks bottle or keg the beer like you would normally do. Remember to clean and sanitize the bottles well. Sanitizing the bottle caps won't hurt either.
Hopefully these instructions will be of use to you. Good luck.
And if you have a bucket of wort that isn't fermenting, try adding another yeast with the instructions above.