First, gently douse yourself with a bucket full of bath water outside of the tub. This is called "kakeyu" (pronounced "kah-'keh-yu"). There are two reasons this is practiced: 1) to wash off excessive sweat off of your body, 2) to let your body prepare for full soak in the temperature of your bath water. Work your way up from your toes to your hands, then eventually to your torso with two or three scoops of kakeyu.
Slowly enter the tub. Avoid a sudden soak since it can be harmful to your heart. Avoid jumping into the tub too quickly: splashing the tub water to others is considered rude. Submerge up to your waist first if the water temperature is too high, then slowly work your way up your shoulders. Do not alter the water temperature by adding cold water in the tub.
Pay close attention to how long you stay in the tub. It can be harmful to your health to stay in a high temperature water over a long period of time. It can result in what is called, "yuatari" (pronounced yu-'ah-tah-lee). Yuatari symptoms include nausea, increase in heart rate, fainting, unable to walk, and so on. It is highly advised that you step out of the tub and take a breather once in a while when it gets hot or too uncomfortable for you.
An after-shower may be needed depending on the onsen and its mineral composition. Especially for those with a sensitive skin, a quick wash is recommended after an onsen with contents including acid and/or hydrogen sulfide. Also, a very thorough wash and a rince are recommended for onsens with a circulated-spring/purifier.
Yukata is a type of Japanese kimono worn during summer festivals and traditional events. Japanese ryokan and hotels typically use "onsen yukata" as pajamas for the guests.
Ask the hotel concierge for further help if needed.
