As recent newbie to Russian, I recommend you spend $14 on
Pronounce It Perfectly In Russian.
This book and cassette describes, diagrams, and demonstrates the proper tongue position for every sound. This is extremely useful, because the relaxed tongue position for speaking Russian is dramatically different than for speaking English. I'm really glad I got this book when I was first starting out, because it has saved me a lot of effort and frustration. And my teacher was really surprised at how good my pronunciation had become after just a day or two of working with this book.
To get you started, here's the big first step:
In English, we form consonants by bringing the tip of the tongue up to the hard palate (e.g. "T") or the upper teeth (e.g. "Th"). But in Russian, the tip of the tongue usually just rests against the lower teeth (or at the bottom of the upper teeth), and consonants are formed by arching the tongue up to reach the palate. So it's the area on your tongue starting about an inch (2 cm) back from the tip that actually touches the palate and creates most of the consonants. Once you start doing that, the soft consonants come naturally, and the eee sound mixed with the consonant comes naturally.
For example, "
сколько" (how many). For now, force yourself to keep the tip of your tongue against your lower teeth. The S and L sounds are created with the tongue arching up to touch the palate just above and behind the upper teeth. You should immediately notice how your S sound is slightly softer, and how easily you can get that eee sound mixed in with the L.
Next, try "дети" (children). If you pronounce the D by arching your tongue up to the palate, the "ye" sound of the e comes automatically.
Another fun word to practice is "
долларов" (dollars). If you pronounce the L as you do in English, it's impossible to flip the R properly. But if you use the arche of your tongue to make the L, the flipped R can happen naturally. And my experience was, once I found the position of the L that made the flipped R easy, lo and behold, my "O" vowels suddenly sounded Russian too.
Disclaimer: I'm just another beginning student, and not an expert on pronunciation. The above is just what seems to work for me. For best results, try the book, and find a native speaker to guide you.