Getting a herbal immune booster will probably help you a bit. I use Immun-o First.
When I do get sick, if possible I stay home and sleep most of the day. Lots of liquids too. I usually don't take lantus, only my novorapid and just test every 2hours but that's something you should discuss with your endo. I often take herbal chinese cough medicine if it's a cough, and don't really worry about the sugar content. I try to avoid medicines though, and from your question about natural remedies, I'm sure you're open to that! i do tend to take half a package of neocritin if i have a cold though to help me nap. it has a lot of sugar in it though, so i don't recommend a full serving.
If i'm really sick and i'm not sure what to do, I usually call my endo in the morning and ask for advise. otherwise, i treat my illness like a normal person does! if you start to get worse, i would call your endo cuz you might need to go to the ER but it'll be rare. some people test for ketones in your urine, but i haven't done that in years. since you're new to diabetes though, i'd recommend getting a box of the strips though.
I rarely get so sick that I'm out of comission for more than a day though, it's only happened a few times in the last 15years, so i wouldn't get too nervous about it :) just remember to wash your hands, avoid sick people if possible, eat well, get enough rest and you should keep your chances of getting really sick down :)
you're endo will be a great source of knowledge about what you should be doing when you're sick, so i recommend discussing a "illness" plan with them as well :)