Good Advice; here's a bit more detail.
Muscles respond to training input much faster than tendons (which attach muscles to the bones) and ligaments (which attach bones to other bones), roughly on a scale of 6 weeks vs 6 months. This is because the muscles have a vastly greater blood supply and connective tendons & ligaments have very limited blood supply. So, it is especially critical for people who are new to weight training and those have not been in weight training for a long time to watch out for this lag.
Tendons & ligaments also take much longer to recover from injury than do muscles.
Your muscles will show measurable improvements in ability to lift more weight & do more reps within weeks. You can keep pushing it some, but be really slow and careful about it for the first couple years. Do more reps and only slowly build up the peak weights. Push it too fast and you may find yourself on the bench for many weeks with an injured tendon. This will setback your training progress far worse than managing your weight increases.
More generally, in all kinds of training, the key is that your body actually gets stronger during the rest periods after being stimulated by the training input.
Good training input actually breaks down the muscles a little bit (too much and it's just injury). The rebuilding process will rebuild them stronger to handle the input just received. So, you must train to get anything to happen, but if you train too much / rest too little, you will merely be breaking down the muscle faster than it repairs.
Another question is pain. You may often hear "push through the pain", "no pain, no gain", "pain is your friend", etc. This is partially true, but it really depends on type the pain. My general rule of thumb is: the brurning pain in your muscles that gets more intense as you get towards failure is almost always fine, push through it. But any really sharp pain that starts suddenly, in muscles or especially joints -- you are already two seconds too late -- stop the loading as quickly as you can safely do (e.g., drop the weight, but not on yourself or your spotter), in order to minimize damage. Also, immediate ice is your friend here.
Enthusiasm & vigor is great, just be sure your body also gets the time it needs to actually build the strength & fitness you seek. I encourage everyone to pursue their best fitness - enjoy it!
(Source: Fmr international competitor in several sports incl US Ski Team)