I moved my household goods to Mauritius about a year ago and didn't pay any duties to bring them in the country. In fact, I was living in Mauritius for a year before I had my items shipped to be sure I was going to stay here and also to see what I would need and what I could get rid of before shipping.
From what I remember, you are allowed to bring in almost anything reasonable as part of your household goods with no duties being attached. You will need to check what duties may be applied to any alcohol or motor vehicles you may try to import as I did not bring any items of that nature. I do know that there will potentially be duties attached to bringing in those type of items.
Depending on where you are moving from, you can also look to buy new items in your country of origin and claim back the VAT/GST there when you take them out of the country. This is what I did as I was previously living in the UK. I was able to buy new appliances (washing machine, microwave, etc) and various electronics (TV, DVD player, etc) at better prices and better quality than what I found in Mauritius and get a VAT refund from the UK.
Once all of your household items have been brought in to Mauritius, the customs department will require that you file a report with them annually for the first two years to certify that you have not sold any of your items and may even do random spot checks to make sure of it as well. If you do sell any items or get rid of any items you brought in and the customs department discovers this, you may be liable to pay customs duties plus VAT and a 10% penalty as well.
Here are the contact details for the customs department who would be able to answer any specific questions you may have:
Telephone - 206 3400
Fax - 240 1032 and 240 0434
Email - customs@mra.mu
I would highly recommend that you come to Mauritius first and see where you will live and the size of your accommodation and what items you will need. Then try to completely fill up whatever container size you ship your items in as it will not make any difference in the cost of shipping your household goods. This way you can be sure that you have everything you need and if you need to purchase anything such as appliances or electronics, you will more than likely find a larger selection of better quality items to choose from at lower prices in your country of origin.
Regarding bringing pets in to the country, cats and dogs have a minimum quarantine of 30 days if brought from the US, UK and Australia according to my friends that have brought their pets from those countries. In some cases you can get them out a bit early as one of my friends had their cat and dog from Australia released in 25 days. I believe that if the pets are coming from South Africa though, the length of quarantine is 90 days, but I'm not 100% sure.