It takes too long for 1st gear to shift.

Shifting is dependent on accelerator pedal position, engine rpm and vehicle speed and varies with no fixed rpm for upshifting. Why do you think there is a problem?

Sorry for the delay in replying. My phone line had been down for the past 2 weeks due to cable fault and has not been able to get online.

Late shifting might not be a problem but harsh shifting is and I would suggest with a check for trouble codes.

300ZX (1993-94)

1. Ensure engine is at normal operating temperature. Turn ignition switch to OFF position. From OFF position turn ignition switch to ACC position. Set overdrive switch to OFF position. Move selector lever to "D" position. Turn ignition switch to ON position, but DO NOT start engine. If A/T check lamp comes on for about 2 seconds, go to next step. If A/T check lamp does not come on for about 2 seconds, see Fig. 10.

2. Move selector lever to "2" position. Set overdrive switch to ON position and move selector lever to "1" position. Set overdrive switch to OFF position. Depress accelerator pedal to floor and release. A/T check lamp will flash trouble code. Refer to appropriate SELF-DIAGNOSTIC CODE chart. See Fig. 11 (only part of codes are shown as example) Let me know what you have and I will get the exact code explanation for you.

After step 2, nothing happened?
That would mean no trouble codes and you would need to proceed with solenoid testings. You would need a DVOM to test the resistance of the solenoids.