Or, you could have a group of smaller and hardy fish (equally beautiful as the larger fishes are) in the beginning. White cloud mountain minnows, for example, are small and very hard - excellent for beginners (but they're by no means the only fish).
You ask what you need to know to get you started on the right track. Unless you already know about it, first you need to learn about the Nitrogen Cycle:
http://www.fishlore.com/NitrogenCycle.htm. You should also read the rest of the articles:
http://www.fishlore.com/Beginners.htm. After you've read these, you'll know why it's good to have tests for ammonia, nitrite, nitrate, and pH. Perhaps you'll also know why all of your fish have been dying. (What I mean is: If you were not aware of the Nitrogen Cycle before, it was the most likely cause for your fish deaths.)
As for filtration, your filter isn't necessarily "bad" because your fish have been dying. You'll know what I mean after you've read about the cycle. Anyway, what filter do you have in your 10 gallon tank? A good external power filter should pump at least 10 times the volume of your tank - so you want a filter that pumps at least 100 gallons per hour (in your case).
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I'll be deleting my post then...