My answer will always be strength.
Don't get me wrong, cardiovascular conditioning is a huge part of health and performance, whether you are training for a marathon, a Physical Abilities Test (PAT), or are seeking to improve your health. However.... be strong first.
Part of the reason I recommend strength first is because of how long it takes your body to adapt to strength training demands. Initial changes in strength occur quickly, however most of these improvements are your nervous system becoming more efficient. This is usually accompanied by only a small change in body composition (i.e. increased muscle mass).
Even though changes in strength can occur quickly, many people plateau because of poor programming or not properly managing their intensity. (This can be prevented, or at least minimized, with proper progression and programming). This plateau usually occurs about 3 months into strength training.
Other changes associated with strength take much longer, and even up to several years, to develop. While your muscular strength will increase quickly, other adaptations including tendon, ligament and bone adaptations will take much longer to match your increase in strength.
For those of you who are training for a PAT, the bright side is that you can usually reach the level of strength you need to pass before you hit the plateau. But for those of you who are smaller in stature, and for many female Law Enforcement Candidates, it can take as long as 6 months or even a years worth of training. Take for example a woman who is 5'5" and 120 lbs. She is going to need to be able to Bench/Floor Press a much larger percentage of her weight than another who is 5'10" and 180 lbs.
Aerobic conditioning, on the other hand, can be trained very quickly. If you really push yourself (with a HUGE focus on recovery) you can reach an adequate fitness level (whether for health or the PAT) is little over a month of consistent training, starting from a moderate fitness level.
It is very rare that my answer to this question is conditioning. Human beings are wired to be aerobically fit, regardless of body composition. That is one of the reasons we respond so quickly to conditioning training.
Start with strength. You are never wrong when you increase your strength (CAUTION: when you increase your RELATIVE strength).
B.Kin, Canadian Society of Exercise Physiology CPT