In my and others' experiences, the first year on a gluten-free diet can be very challenging. 
T o tackle the challenge, you may want to print out lists of gluten-free groceries sold at the stores where you usually shop. Major chains who carry GF items--like Henry's, Trader Joe's, and Whole Foods Market, will include these lists on their web sites. But if you shop at, say, a Kroger-affiliated grocery store, you'll have to make your own list.
If you're at all like me and my celiac friends, you'll have to spend more time in the kitchen and eat a fresher, more nutritious diet. 
Added kitchen time is not only necessary to save on your grocery bills because major names in processed, gluten-free foods are hard to come by; but also because the diet you'll give yourself will help support the nutritional deficiencies you likely have. (Celiac is classified, both as an autoimmune and a malabsorptive disease.)
Major names at big-box grocery stores offer few gluten-free items--especially ready-to-eat processed foods. Heinz is one major name that--assuming you tolerate cow's milk--makes eight frozen, dinners you can safely eat. These are "Weight Watchers Smart Ones" dinners: Santa Fe Rice & Beans, Broccoli & Cheddar Potatoes, Lemon Herb Chicken Piccata, Fiesta Chicken, Creamy Tuscan Chicken, Grilled Chicken in Garlic Herb Sauce, and Home-Style Chicken Chicken Santa Fe. (http://www.heinz.com/health--wellness/focus-foods/dietary-preferences/gluten-free.aspx) And in the last couple of years, Rice and Corn Chex removed all the gluten from their breakfast cereals. Virtually all other big-name breakfast cereals do contain gluten, as do most seasoned snacks (chips, nuts, etc.), among many other processed foods, yet to be labeled under a two-year-old federal law! 
Whether having wheat gluten or barley malt, about 95% of all processed foods are unsafe to celiac patients like me and you. With time, I've found it easier and easier to live with this fact, as I believe you will, too. 
Although nearly unimaginable a couple of years ago, this is now part of my experience, even though I'd had severe GI pain, migraines, and cardiomyopathy threatening my life in 2007. Some reflections I'd like to share with you at this point in my life are: The gluten-free diet is a slow, but sure road to better health for virtually all celiac patients, even though we may become less and less conscious of improvement in our health as the worst symptoms subside at first. It's an eery, upsetting yet thrilling and uplifting experience to have eaten a poisonous food (wheat bread) at almost every meal, three times a day for the period of your life up to the time of your diagnosis and then completely eliminate it from your life. One last reflection is that there are many great gluten-free dishes available nowadays (unlike just a few yrs. ago and certainly unlike a few decades ago), but it simply takes more planning to shop and go out to eat. This makes eating an inconvenience but also a personally rewarding experience! 
To respond your question, cross-contamination due to wheat being in the same facility is not usually a problem, and should not be so if the product is labeled gluten-free or has a statement that strict manufacturing practices are in place to prevent any contamination. On the other hand, a statement appearing on the packaging of a product that it was made on the same equipment as wheat will always lead me to the decision that it's unsafe to eat, unless it's labeled gluten-free (usually in addition to the statement about such mfg. practices). As someone who likes to help others, especially those who've walked along the same path as I have, I wish I could offer clearer advice. But the next piece of advice I would offer is to learn what brands and what items are reliable. In other words, wheat (or gluten thereof) is contained if and only if it's included on the ingredients list. I believe the ingredients lists of Heinz to be so. As for spices and seasoning, I believe Lawry's, McCormick, and Sun-Bird brands also have such a policy regarding their ingredients lists. However, you'll want to do your own research and read every label every time. My wife and I have made it standard operating procedure in our kitchen to read each label once at the grocery store and a second time before unpackaging the processed-food item. Stay vigilant! 
Do keep in touch with us here on line. And if a celiac support group meets in your area, join it for all the moral support you need and deserve! 