A tick should never be covered with oil or ointment. This causes it to choke and expel its gut contents into the bite wound, increasing the risk of infection transmission. The tick should also never be squeezed by hand or tweezers. The safest way to remove a tick is using a special tick-card with a small notch to pull the tick out of the skin without the risk of squeezing the tick and forcing its contents into the bite wound, thereby minimizing the chance of infection. If dealing with a very small tick larva, a dampened cotton swab or a matchstick can be used to wobble the tick out of the skin. Disinfect the bite wound properly with a regular disinfectant. If the tick is located at a badly accessible place, it is possible to use a hook with a notch for the removal. If none of the methods described work, sharp tweezers can be used to carefully clutch the tick as close to the skin as possible and pull it out.

Detailed instruction on how to remove a tick using the tick-card can be found in the linked article.

The use of latex gloves is not necessary if you do not have raw skin abrasions or bruises. It is however advised to wash your hands thoroughly before and after the extraction process.

If the ticks feeding tube (hypostome) stays in the skin, there is no need to panic. The tube itself is not a source of infection and the healing skin will eliminate it on its own. The bite wound needs to be disinfected and monitored for signs of inflammation.
A reddened bump can appear at the bite area. This is only a local reaction which will subside in time. Reddening, edema or itchiness of the bite area, therefore, does not indicate a transmitted infection.

If you wish to have the removed tick analyzed for possible disease causative agents, do not throw it away. Such an analysis can be a useful tool in finding out if the tick is a disease carrier and consequently applying preventive measures to prevent health complications. Insert the tick into a sealable plastic bag and send it to our laboratory for analysis. The use of disinfection, soap or ointments does not interfere with the analysis.

Information about the analysis can be found in this link.

For a safe disposal, flush the removed tick into the toilet.