Different projects require different needles. Generally it is handy to have a lot of universal needles on hand, for general projects. Thicker needles (up to American size 19) are better for thicker fabrics and threads, while smaller needles (down to American size 8) are better for thinner fabrics and threads. DenverFabrics.com recommends that you thread your needle and, holding the thread taught, spin the needle down along the thread. If it slides down, you are good, if not, you may need a larger needle.
If the universal needle is skipping stitches on you knit fabrics, then you will want to use a ballpoint needle, which pushes through the threads of the fabric without piercing it.
Similarly, you may want to use a sharp/microtex needle for silks and other fabrics which require being pierced by a sharp needle.
You can explore more specialty needles at DenverFabrics.com.
Note: Whatever needle you use, make sure to replace it every few projects. A dull/used needle is the main cause of most stitching problems.
For any project, I recommend European thread brands like Mettler or Gutermann, which are of better quality than brands like Coats and Clark. As the Colette Patterns blog notes, cheap threads, which can break or get stuck in the machine, can ruin a project.
For a list of ten basic mistakes that can ruin your sewing project, check out the full Colette blog post.
Once you have figured out the size of the pattern piece you will make (see this post on getting the right fit for your body), use a highlighter to mark the cutting lines on a multi-size pattern. This is especially useful if you are different sizes, such as a small at the bust but a medium at the waist. Even if you are a single size, the lines can get especially confusing around armholes and other details, and I have cut along the wrong line far too many times to risk it again.
To mark elements like darts and other marks, with wrong sides together pin through each dot on the pattern, open up the two layers of fabric, and mark the dimply on the wrong side of the fabric that is left by the pin. You can use Mark-B-Gone (spritz with water and iron to disappear; doesn't work on polyester), Disappearing Ink (disappears within 24 hours; not good for long-term projects), tailor's (or regular) chalk (be careful: waxy chalk can stain fabric), or tracing paper (make sure to also get a tracing wheel). Make sure to always test your markings on a scrap piece of fabric to make sure it works for your fabric!
Make sure to use sharp fabric scissors. These should be kept separate from all your other household and sewing scissors, and only used to cut fabric to keep them from becoming dull. In order to achieve smoother, less jagged cuts, take long cuts instead of short snips and avoid turning your had at awkward angles (turn the fabric or moving around your piece instead of becoming a contortionist). Scissors with angled handles can also help you cut smoothly across the fabric, again because they make it easy to hold your hand straight against a flat cutting service rather than lifting the fabric off the table to cut.
For tips on finishing your seams, check out the Seam Guide.