It's common for beginners to experience muscle soreness that lasts for a week or two, just as seasoned exercisers will be sore after a tough workout. Yes, you should keep working out even though you are sore, but there is more to it than that.
Muscle soreness has two primary causes. The first soreness you experience happens during your workout ("the burn") and should subside within a couple of hours. This is caused by lactic acid production. When you are training and your muscles are not getting enough oxygen (anaerobic glycolysis), lactic acid builds up. You can break down lactic acid by continuing to move and by doing light aerobic exercise (such as walking) after your workout. This is why cool-downs are so important, especially for beginners. The longer you cool down, the faster that lactic acid will leave the muscles (typically within an hour).
The type of muscle soreness you are experiencing, up to a day or two (and sometimes even three) after your workout is known as DOMS (Delayed Onset Muscle Soreness). DOMS is caused by microscopic tears inside the muscles, resulting from weight-training or fully exhausting the muscles during cardio. This is normal. Again, beginners will be more sore and usually for longer, but if you really worked as hard as you should have during a weight-lifting session, you should be somewhat sore for the next day or two.
This is where rest comes in. You absolutely must rest the muscles you worked for 1-2 days after a workout. Take at least one day off between strength training sessions, and if you are still very sore, take 2 days off. (This means from lifting, not from all exercise such as cardio). If you don't let your muscles recover and repair, they will continue to break down and you will actually get weaker.
To help prevent soreness in the future, and alleviate some of it now, be sure to:
1. Always warm up for 5-10 minutes and cool down for at least 5 minutes.
2. Stretch after a warm-up, during your workout, and after you are done. Only stretch when your muscles are already warm from some kind of light activity.
3. Stay active. The more your muscles move, the faster they will recover from exercise and soreness. If you choose to rest completely instead of "actively recovering" with light exercise, you'll probably be sore longer.