As requested, here are some tips and general guidance for all models of Apple Pencil. Hopefully you'll find this information to be useful if you experience any difficulties:
1) The Apple Pencil tip is a consumable item - that should be replaced as soon as any wear becomes visible. Never use a Pencil whose tip has worn to the extent that any part of the metal transducer is visible - as you may damage both the pencil and iPad screen.
2) Ensure that the user-replaceable Pencil tip is snugly screwed into the body of the Apple Pencil; when correctly fitted you should see a tiny gap, of about the thickness of a sheet of paper, between the tip and pencil body. If the tip is slightly loosened, or overtightened, the Pencil may not work properly, if at all.
3) While Pencils are relatively robust, treat them with care. The transducer element (that sits immediately under the replaceable tip) can suffer damage from impact or excessive lateral force to the tip - such as might occur if the Pencil is dropped onto a hard surface. If the transducer beneath the plastic tip is distorted or damaged, the Pencil will require replacement. Repair of an Apple Pencil is not possible.
4) Be aware that if a Pencil has been set-aside for an extended period, unused and uncharged, the tiny Pencil battery can be damaged through deep discharge. In common with all Li-ion batteries, they must always have some level of charge.
The tiny internal Li-ion battery is susceptible to permanent/irreversible damage through being left discharged for long periods. Even some “new” pencils can exhibit signs of failure out-of-the-box if they are “old stock”.
It is essential that if you have an Apple Pencil that you charge it regularly - whether used or not - so as to protect the battery from deep-discharge. Do not allow a pencil to remain in low-charge state for any period of time - as the internal battery will fail, rendering the Pencil useless. Setting aside an unused Pencil, for extended periods, is a recipe for premature death of the Pencil battery.
Gentle warming of the Pencil (such as placing on a hot water radiator), prior to attempting charging of the Pencil, can sometimes resurrect a dead Pencil battery. If the Pencil battery fails, the only remedy is to replace the entire Pencil; the battery is not a serviceable item.
5) The second generation Apple Pencil and Apple Pencil Pro are charged by the host iPad using inductive charging. It is generally recommended that these Pencils should be kept docked with the host iPad when not in active use - as this ensures that the tiny internal Pencil battery is kept optimally charged when not in use. Charging will only occur when the battery requires charge from the iPad, optimal charge being entirely automatic.
Keeping the Pencil separated from the iPad when unused risks allowing its battery to fully discharge and suffer irreversible damage. An otherwise healthy Apple Pencil will suffer no negative consequences from being docked with its host iPad for extended periods.
6) If an Apple Pencil fails within its one-year warranty, you should look to having it replaced by your retailer or at an Apple Store.
Genius Bar Reservation and Apple Support Options - Apple
7) General Troubleshooting:
These support pages may be helpful with pairing and use of the Pencil with your iPad:
Connect Apple Pencil with your iPad - Apple Support
Charge your Apple Pencil and check the battery - Apple Support
If you experience difficulties, here is one of the better third-party troubleshooting guides:
https://appletoolbox.com/apple-pencil-not-working-heres-our-troubleshooting-guide/
