Hi Minette and in advance welcome to Ras Al Khaimah 
The majority of expatriate community lives in Al Hamra Village or Mina Al Arab both are located somewhat 25 km outside the city but there are shopping facilities around such as Al Hamra Mall with a Spinney's supermarket, some fashion shops, shoe shops etc as well as a small food court. They are still building an extension which will house a Vox cinema complex supposedly opening by the end of the year.
In Mina Al Arab, there will be a Choitram in the near future as well as a complete spa with hair salon for gents/women.
On the main road to RAK there are also several other supermarkets.
Regarding schools, there are several possibilities :
RAK Academy Three Schools, Two Campuses, One Academy.
Ras Al Khaimah Academy is one of the oldest, largest and most prestigious international schools in the UAE.
The school was originally founded in 1975 and opening under the name RAK English Speaking School RAKESS with just 20 primary- school, expatriate students on January 17th 1976.
For the next 16 years the total number of students in the school hovered around the 100 mark.
As the schoolÂ’s reputation grew, so interest in the school grew from within the local Emirati population.
The Secondary School opened in 1994 and the school enrolment as a whole began to grow at almost 20% per year.
The prestigious programmes of the International Baccalaureate (IB Diploma and Primary Years Programme) were added in 2006 and a British Curriculum Primary School opened in 2007. Today RAK Academy welcomes over 2,800 students every day from over 60 nationalities. For more information :
I must add that most expatriates have enrolled their kids in that school.
GEMS which you cited has opened their doors only in September but the reviews I've had during the past weeks are not very encouraging unfortunately namely by the way they were treated when going to the school to get information. Also GEMS have different quality levels in UAE and apparently the one in RAK is not one of their best and they are known in UAE to regularly raise the fees without any reason.
Of course, it's up to you to decide where you will enroll your kids 
For the youngest one, a new nursery/pre-school establishment opened its doors right outside between AH Village and Mina Al Arab. I personally went to visit and can say they are very professional and have created a very nice environment for the kids.
I hope to have answered your questions but please do not hesitate to contact again if you need more info 
Arlette