You have to click the switch twice because the first click turns on the low-wattage filament of the "3 way" bulb - which doesn't exist in the ordinary bulb you installed.

The single filament in a regular bulb is on the center contact of the bulb.

The second filament is on the ring contact on the base of the bulb.

When only one filament is present the ring-contact is not doing anything.

You can replace the socket with a regular one if you do not want to use 3-way bulbs.

The second contact in the 3-way socketys has not been manufactured correctly for a number of years.

The Chinese seem to have gotten the idea of the contact, but have failed to provide and spring action making them not work well.

The switch contacts themselves have also gotten very poor, often with short life.

But is is all OK since they have made them so cheap.
I just get tired of replacing them every 6 months.

Thanks for the suggestions. I want to use the 3-way bulb, but it won't work. I thought maybe I was missing something.

If you have another lamp fixture, that also uses a 3-way light bulb that is working properly, try that bulb in your new table lamp.

As brickeyee stated, quite often the sockets are not manufactured as well as they should be. Sometimes the light bulb may need a little bit more (carefully applied) tightening to make contact and work properly.

If that does not work, check the bulb again in the lamp fixture you removed it from. If it works properly, you have confirmed you new table lamp is the problem and should be returned.

Also, you may have a broken filament in the 3-way bulb. Does it work properly in another 3-way lamp?

Thanks for all the help. Tried the bulb in another 3-way and it did not work there either, so am buying a new one. If that doesn't work, guess I will use a single watt.
Thanks again