Price will determine the type and quality of drum kit that you can buy. If you are working on a tight budget finding a used drum set could be a good option. When you buy used you can usually find nicer drums for a cheaper price. Check places like Craigslist for used gear.
Consider what style of music you are interested in and will most often play. Every style of music has a signature drum sound. For instance, jazz drummers tend to have drums with higher tones; rock drummers tend to have drums that produce lower tones with more punch to help fill out the big "rock band" feel. If you want to play all styles of music then maple drums might be a good option since they are somewhere in the middle of the tone spectrum of drums.
The type of wood that a drum set is made of affects the tone of the drums. Softer woods produce lower tones with less projection. Harder woods produce higher tones with more projection. There are three main types of wood that most drums are made out of: maple, birch, and mahogany. Mahogany is the softest of the three and will produce lower tones and has a great bottom-end punch. Maple is somewhere in the middle, and tends to be the most popular. Birch is the hardest wood of the three and produces a higher tone with a lot of projection. A lot of cheaper drums are made out of poplar—the least desirable wood for drums since it is basically particle board.
Size Does Matter
The size of the drums will affect the pitch. A 16" drum will have a much lower pitch than a 10" drum. Think through what style of music you most play, this can help determine what size drums you buy. Rock drummers may want larger drums that produce lower pitch, and jazz drummers may want to buy smaller drums with a higher pitch. Another consideration is the size of the player. I'm often asked, "If my son/daughter is too small for a “normal-sized” drum set, what type of drums should I get?" A great question. What I normally recommend is the Gretsch Catalina Club drum set. It is a smaller drum set that will fit a smaller player—but still is a great long-term purchase. I would avoid the "Walmart special" drums, unless you suspect that playing the drums is a phase that will pass.
One large factor to consider when choosing drums is the venue that the drums will be most utilized. Is your drum kit going to be most played in an apartment, house, church, arena, or small club? You might want to avoid the Neil Peart extra-big-and-loud drums if you’re going to mainly play your drums in an apartment. A smaller and less projected kit would be a good consideration for smaller venues. An electric kit is a good option for those who live in apartments and want to practice late hours in the night.
The last and most important tip is to use your ears. Which drums sound good to you? This is a long-term investment and you want to love the drums you get. Play the drums that you are interested in and decide if you like the sound.