The first point is very important for avoiding altitude sickness and your guides will likely keep reminding you:
Keep drinking! It’s very easy to dehydrate at altitude without noticing. The air is very dry so you breathe off more moisture. Also, your body adjusts to the high altitude by eliminating more water. Keep replacing it.
Also, make sure you eat plenty! Most people lose their appetite at altitude, but the cold weather and the long days mean your body burns through a lot of calories. Keep replacing them. You will need them. High carbohydrate foods are better than fatty foods. (Any good tour operator will have considered that in their shopping and meal planning.)
And keep warm! The correct gear is a must, not just because shivering isn’t nice and hypothermia dangerous, but also because staying nice and toasty will lessen your risk of succumbing to altitude sickness.
Keep your day pack light. Only take what you really need. Every extra kilo needs extra oxygen to carry.
And last but not least, avoid alcohol, tobacco, and most definitely do not touch sleeping tablets! Or you may not wake up again…
And that’s about it. Even if you are not in a position to afford extra preparation for the altitude (e.g. a Mt. Meru climb), if you are healthy, pick a good route and operator, arrive a couple of days early and take on board all of the above tips, you have a very good chance of making it to the summit.