Why would your car make a ticking noise when you start and accelerate?

If the unit ticks at idle and under acceleration then you have to do some testing. I suggest the following:

Check the oil level first. If the oil is old change it. If that clears up the ticking then you've been too long between changes. Bolt up an oil pressure gauge to determine if low oil pressure is the cause. Oil pressure is typically spec'd by the manufacturer and if the pressure is low then one has to assume the oil pump is failing.

If the oil pressure is good and the noise is still present use a plastic tube as a "stethescope" to localize the noise.

Low on the engine indicates main or rod bearings. Mains generally do not tick, they more likely thunk or klunk so bet on rod bearings.

High on the engine (head, valve cover area) indicate valve train noise. In a newer engine it's probably a worn/broken hydraulic lifter. Time to pull the cover and inspect the CAM (if OverHead Cam) and lifters. If you determine that the engine is worth the repair secure a set of lifters and install them.

If the noise occurs only when rolling then determine if the noise changes (gets louder) on left and right 90 degree turns. If this is the case then rebuilt axles or CVs may be in order.

If you go the repair route buy the manual for the car. I recommend Haynes or Petersens, I have found Chilton's manuals to be wanting.