I often feel the need to pop my fingers, elbows, jaw, neck, legs, and even spots by my ribs and pelvis. What type of doctor should I see that can help me figure out what's wrong with me?
Some people simply have "looser" joints throughout their body due to less elastic connective tissue (and other causes), a condition called benign joint hypermobility. It sometimes manifests itself as a pervasive vague joint discomfort or dull soreness which may be temporarily relieved by popping or stretching the joint. The joints may also pop spontaneously such as when turning your neck or twisting your torso. In and of itself, this does not necessarily mean there is anything wrong with you as long as you're not having frequent pain, spasms, swelling, or instability of the joints. Sometimes people get so used to popping their joints to relieve that discomfort that they almost get "addicted" to it, doing it multiple times a day, which can actually start to cause inflammation in the joints. Once they limit how often they are popping their joints the worsening discomfort usually subsides, possibly aided by a short course of over-the-counter anti-inflammatory medicine (e.g., naproxen or ibuprofen.) If you are having worse symptoms than that, it may be a different condition that does warrant being checked out by a physician.
If you do decide to get it checked out, you should probably start by seeing your primary care provider (typically a family practice or internal medicine physician, if you don't have an established medical provider.) They can evaluate your full medical history and decide if any testing, treatment, or referrals are needed. Ultimately, the medical specialist you may need to see is a rheumatologist. They diagnose and treat conditions which tend to affect multiple joints and connective tissues throughout the body, such as rheumatoid arthritis and lupus.
Regarding the joint hypermobility, this link contained some useful information: http://www.healthline.com/health/hypermobile-joints#Overview1
One further thought about joint hypermobility is that strengthening the muscles around the bothersome joints can make up for the joint capsule and ligament laxity and stabilize the joints, thereby lessening the symptoms.