Well, if your animal products are getting to a very low amount, you can consider taking a B12 tablet. If you become a strict vegan, it is certainly reccomended.
You need a source of Omega 3 which can be chia seed, flax seed or walnuts.
Iodine is a tricky one. You can check if there is iodine in the salt your family uses. I also eat cranberries and strawberries for iodine. It can be low on a vegan diet and many articles you will read miss this one out, but it is absolutely in the top three important things vegans are most likely to be low in.
The above three are perhaps the only three things which can be particularly deficient in a vegetarian or vegan diet. If you eat a well balanced diet to include legumes, grains, vegetables (including green veg) and fruit then everything else should take care of itself.
Maybe calcium is one to think about - almonds,orange, bread, broccoli are some example sources, but it does help to have some food (such as vegan milk) fortified with calcium as it can be low on a vegan diet.
Apart from the above advice, assuming you are healthy and have no particular allergies/conditions etc that you know of, there isn't really any need to go much deeper than that.
Also, there is no urgent need to sort all this out instantly, you can work the above things steadily into your diet over the next few weeks perhaps, or months even.
If you want to look into nutrition in more depth, you can look at
http://www.veganhealth.org/ and its articles, it is well respected. However, don't think you don't need to understand nutrition in huge depth to be a healthy vegan. It helps but I think just reading the advice in my post and following it covers the basics already to make you healthier than the average meat eater.
Let me know if there's anything you need help with right now. I will try and help you if I remember to come back to this thread.