Make sure that you wash and then thoroughly rinse your hands and arms before putting them into the tank. Better yet, use rubber gloves which are rinsed off after each use - I don't do this, but it's still recommended :-)
Look at your tank. This may seem like a silly suggestion, but the best way to keep your tank healthy is to become familiar with the way everything looks when they're happy so that you can quickly tell when they're not happy.
About once a week, or slightly more often if you have the time, I clean the algae that grows on the inside of the glass. I've found the best thing to use for this is an old credit card. The fibrous sponges pick up sand from the bottom too easily, and these sand granules can easily scratch a Plexiglas tank. The time required for this cleaning is dependent on how long its been since I did it last. I can usually do this in 10-15 minutes if I'm doing a real careful job, or 5-10 minutes if I'm in a hurry. The more often you do this, the easier and faster it is, but it's not a good idea to put your hands in the tank any more often than necessary.
Top off the water that has evaporated with filtered fresh water. Don't add more salt water since only the fresh water have evaporated. Make sure you use a good carbon based or reverse osmosis filter to remove impurities such as heavy metals and phosphates. This will reduce the algae growth and make your reef life much happier.
You can consider adding replenishment chemicals such as strontium, magnesium and iodide. These are available in fish stores. I generally don't do this weekly, but if you can it is preferable. Make sure that you monitor chemical levels periodically at least for the first couple months to make sure you know how much of these chemicals are being consumed in your tank. Too much of any chemical is as too little.
You should not have to replenish calcium if you use the Jaubert system described in Tank Setup, but you should check the levels occasionally just to make sure (monthly is probably adequate).
Empty the protein skimmer waste reservoir if it needs it. I've noticed that the amount of waste that is generated varies quite significantly from week to week. It is generally quite low, but occasionally will be higher.
Approximately once a month, you should check all chemical levels to make sure they are reasonable. Once you've become comfortable observing the tank, you will probably notice a problem way before you would get around to performing a chemical check. The chemical check is a good way to figure out what the problem is (or more likely, what it isn't) when things don't look happy, however.
If you have a lot of detritus (waste products - looks like brown powdery material) in the tank, you can use a small powerhead to blow the detritus off the live rock and into the water. The mechanical filter in your sump will clear out the waste. You may also need to siphon out the detritus that has accumulated in the sump. Replace the siphoned water with new salt water at the appropriate salinity.
Check the salinity of the water. The salinity can go down if salt crystals accumulate on tank covers, the side of the sump, etc. If it is low, increase it very slowly to the right level by adding salt dissolved in water (at a higher salinity) to the sump.
Change the mechanical filter material (I use sheet filter floss). Wait until after the tank has cleared if you use a powerhead to blow the detritus off the rocks in the tank.
Clean the sump, protein skimmer, and powerheads. Deposits can build up which reduce the effectiveness of these devices. After cleaning, make sure the powerheads appear to be operating at rated capacity. If not, it may be time to replace them.
Replace the florescent light bulbs. These will lose brightness over time. It is actually much better to replace only one at a time since this will minimize the light shock that will result if you change four to six worn bulbs with new ones all at once. If you change one every two months, you should have good results. Mark the date using a permanent marker on the top side of the bulbs so that you can tell when it was installed.
This is probably obvious, but if a coral falls down, you should put it back in place as soon as possible. After a while, most corals will fuse to the live rock on which they are resting and will then stay in place. As long as you don't have any big creatures that like to push things around, you shouldn't have too much problem with corals falling. Some books recommend that corals be epoxied in place. I haven't found the need to do this, and this reduces flexibility if you purchase a new coral that necessitates a little tank rearranging.
In the wild, reef organisms feed at least once a day. Most books recommend a small amount of food once or twice a day. If you can do this, the fish will probably be happiest. But once you have an established tank, your tank will be close to self sustaining because it will create a more-or-less complete food chain and daily feeding will become less important. However, even if you have relatively few fish in your tank (one 1-2" fish per 20 gallons), this still represents a higher density than on the reef so some supplemental feeding is required.
I use frozen Brine Shrimp Plus which comes in small cubes and includes vegetable matter in addition to brine shrimp. For my tank, which includes five fish (~2" each), two brittle stars, two cleaner shrimp, and a bubble tipped anemone (as well as numerous corals), I use 2-3 cubes every 2-3 days. I don't know if this is too much or too little, but with this feeding regimen everything has grown and seems healthy. The anemone grew so much that after about 12 months it split into three anemone, all of which were at least 50% larger than the original anemone I started with.
Thaw the frozen cubes in a cup of water, and break them up into fine pieces with a fork (after thawing) before dumping into the fish tank.
Every few months, I siphon the detritus out of the sump and replace this water, which amounts to a 5-10% water change. I don't do water changes for the sake of changing the water. As long as the water that is replaced due to evaporation is well filtered, it does not build up heavy metals (which is one of the reasons water changes might be required). The Jaubert system described in Tank Setup keeps the nitrate and nitrite levels to a minimum.
Again, many books recommend periodic water changes of a significant percentage of the tank water. But I believe this results in a huge shock to the organisms in the tank. If you can keep the water clean through beneficial bacteria and other organisms, mechanical filtration, and protein skimming, I believe you are better off not changing significant amounts of the water. If you have the time, it probably wouldn't hurt to change a small amount (no more than 5-10%) more regularly but you must be careful to not make any significant changes to salinity, pH, or temperature when doing this.