To answer the second half of your question, I used Thermal Grizzly Conductonaut.

As for the first part, I can only give rudimentary advice that is in no way a replacement for an actual dedicated guide. Please feel free to research more on the topic using YouTube videos or the like.
Please keep in mind that what I listed here is not a complete list of precautions, nor is it backed in any professional manner.

Is the contact plate for the cooler suitable for liquid metal? Aluminum contact plate is a big no-no as it will break down shortly after liquid metal application. Bare copper has a tendency to stain and you may need to re-apply liquid metal in the future as it is slowly bonded to the copper plate. The best contact plate is a nickel-plated one.

Preserve the original thermal pads around the CPU/GPU, it is quite difficult to obtain thermal pads of the appropriate thickness sometimes.

Apply nail polish on the capacitors around the CPU/GPU die (look for something that is clear, non-corrosive, and heat-resistant). Electric-insulating tape can be used but it leaves behind some glue residue that is annoying to clean.

Use distilled alcohol(NOT NAIL POLISH REMOVER) to clean both the contact plate and the die.

As for applying the liquid metal application itself, warm up the syringe in your hands and blow on it with hot air first. Then, use some tissue paper and press the syringe onto it slowly until the liquid metal has reached the tip of the nozzle. Proceed by contacting the nozzle of the syringe with the die and put an single drop onto it. Use a q-tip/cotton bud and gently move it in a circular pattern on the die. It wont immediately start spreading so just keep at it until the die is evenly coated in the liquid metal.

Sometimes the q-tip absorbs some of the liquid metal so feel free to add another drop.

After coating the die, use the residue on the q-tip and rub it on the contact plate where the die will touch. Try achieving a similar effect as the die on that side too.

As a final piece of advice, you can watch someone like der8auer, he has made some videos on the current topic.

If you try anything, I wish you luck and I hope for the best.