The short answer
No. Soreness (DOMS) mostly appears when you do something new or unusual, especially long lowering phases. It fades as your body gets used to the exercises, while you keep progressing. A better sign of good training: your reps or loads go up over the weeks.
What soreness is
Delayed-onset muscle soreness appears 12 to 72 hours after a session. It is linked to small muscle damage and inflammation, especially after eccentric work (the lowering phase) and new exercises.
Why it is not a good indicator
Schoenfeld and Contreras (2013) point out that soreness and muscle growth are not closely linked: some muscles rarely get sore, and soreness fades with repetition even though gains continue.
The real signs of progress
More reps at the same weight, more weight for the same reps, cleaner technique, measurements or photos that change over the months. That is what a workout log is for.
Training while sore
Light soreness: you can train, and the warm-up often makes it fade. Strong soreness that limits your movement: train another muscle group or lower the load. A sharp pain or a joint problem is not soreness.
Limiting it
Increase volume gradually, especially when you start or come back after a break. Sleep, protein and light activity such as walking or cycling help you feel better; no method makes it disappear completely.
Put it into practice
About this guide
STARK editorial guide. Examples explain how to use the library and record training; they are not individual coaching.
Training principles: Schoenfeld & Contreras (2013), Strength and Conditioning Journal.
The session examples come from the STARK program catalogue linked above.