For starters, great advice Jeff...
I have a 12 lb, 4'+ Arg Tegu that is the biggest sweety in the world...
It's very common for Tegus to be flighty when they first arrive. Within the first 1~2 weeks they usually relax a lot. There are quite a few 'tricks' to helping them get used to being handled, but it sounds like yours is well on it's way.
I believe your Tegu closing his eyes while being handled is a sign of comfortability, not stress.
I use straight Cypress Mulch as substrate for my tegu. It holds moisture very well, is quite resistant to mold/mildew, is inexpensive (where available), is easy to manage, etc, etc... I know of others who use a soil mulch mix and are happy with the results, but it's not something I would do. If for no other reason it seems the Tegu would be dirty and then when handled or free roaming would get me or the carpet dirty.
I think the Temps Jeff quoted above are ideal for Columbian tegus, but the Argentina Tegus are from a bit cooler region. I keep my (8' x 3' @ 3') enclosure approx 86*F during the day and 78~80*F at night. He also has a basking area that is around 95~100*F. The basking light is on 8 hours in summer and 4 hours in the spring/fall (off in winter).
The humidity requirements Jeff quoted above are great ideals, but Arg tegus are less picky abotu humidity than Columbians. Too dry of an environment does make shedding difficult and can allow toes to become brittle. But I find keeping the mulch moist works wonders even when the air is dry. If you are having problems or concerns regarding clean sheds, try adding fish oil (Cod Liver Oil) to their meals (easily mixes with Ground Turkey) a couple of times per week. This helps tremendously.
I also have a 6' x 12' outdoor enclosure my Tegu spends time in. Here in North Carolina the daytime temps are in the mid 90s for 3~4 months out of the year. I find my Tegu loves to bask in early to mid morning sunlight before the temp gets past the upper 80s, but once it reaches 90*F he looks for a dark cool place to hide until evening. Once the day cools off he usually comes back out for a while.
Keep in mind Argentinian Tegus hibernate in their native environment. In captivity some do, some don't, but just about all of them slow down considerably in the winter months. Personally I like to adjust their heat/light to reflect seasons, although this is not required. But even when kept at consistent temperatures they still reflect higher activity in summer and lower activity in winter.
You can attempt to prevent hibernation if you wish, but once a tegu decides it's going to sleep for the winter there is nothing you can do to stop it (besides physically wake it up daily which will excessively stress the animal).
I would feed your hatchling daily as much as it will eat. They grow tremendously fast for the first 18~24 months of their lives and they will utilize the energy gained from excessive eating. Adult Tegus are prone to obesity, but young ones aren't.
Jeff gave a great list of foods to offer. Mine highly prefers whole foods such as frozen/thawed mice & chicks and live fish. But I seek to give him a balanced diet to include Ground Turkey, Beef Liver, Crustaceans, Chicken/Turkey Gizzards, Prepared Fish and/or any other raw meats I find at a reasonable cost. I also feed mind scrambled or soft boiled eggs (He commonly gets omlette leftovers on Sundays). When they are smaller (like yours) insects are a wonderful addition to their diet.
Be prepared for this guy to grow like a dang weed. Mine was 6 weeks old at 9" long when I got him in early August '08. He was 16" less than 3 months later when he went into deep hibernation... he woke up in early May at 18" and blew up to over 36" by fall. He was quite literally growing 1.5" per week (yes week!) for many weeks his second summer ('09). This past summer, his third summer, his growth was just as impressive but it was more focused on girth than length.
What size cage do you have your little guy in now? And what enclosure plans do you have for the future?
PS - Sorry for the long post. I've been a little obsesed with Tegus for the last few years *twitch*
