Certainly, visiting Dammam, Riyadh, and Jeddah in Saudi Arabia will be a fantastic cultural adventure. Here are some travel tips to keep in mind:

Best Time to Visit:
The ideal time to visit Saudi Arabia is either during the cooler months, November to February, or during the spring, from March to April and October.

Route and Accessibility:
There are direct international flights to Riyadh and Jeddah. You can take domestic flights or drive to Dammam from there. Saudi Arabia has well-connected highways, but self-driving is recommended only if you are familiar with local customs.

Altitude and Acclimatization:
The altitude isn't a significant problem in these cities as all are near sea level. However, Riyadh is slightly elevated, about 600 meters above sea level. Acclimatization should not be a significant issue though moderate levels of physical fitness will help you adapt better.

Accommodation and Booking:
Saudi Arabia boasts a wide range of accommodation options ranging from luxury hotels to more modest lodgings. It is recommended to book your accommodation in advance, notably during Hajj and Umrah seasons when hotels are often fully booked.

Clothing and Essentials:
Remember, Saudi Arabia has a strict dress code, especially for women who are expected to wear an Abaya (long black cloak) in public. Men should wear long pants and avoid sleeveless shirts. Carry enough sunscreen, as the sun can be quite harsh.

Health and Safety:
Carry essential medicines as well as a first aid kit. The hygiene standards are relatively good, but it's recommended to drink only bottled water.

Local Etiquette and Respect:
Saudi Arabia is an Islamic country, and it's vital to respect the local customs, traditions, and laws. During prayer hours, for example, shops will close for about 30 minutes, and it's important to be patient and respectful during these times. Photographic people without their permission is frowned upon and can have legal repercussions. Always ask for consent before taking photos.