So I was able to figure this out.
I was calling the wrong classes. Web and User where the ones i needed, not SPUser and SPWeb.
I also needed to execute the query to fill up web and current user. Here is a snippit of my code to create a dictionary of some variables.
using (var clientContext = TokenHelper.GetClientContextWithContextToken(hostWeb, contextToken, Request.Url.Authority)) {
Web web = clientContext.Web;
User currentUser = clientContext.Web.CurrentUser;
clientContext.Load(web); //, web => web.Title);
clientContext.Load(currentUser);
clientContext.ExecuteQuery();
//get timestamp
string ts = DateTime.Now.ToString("yyyyMMddHHmmssfff");
Dictionary<string, string> dictionary = new Dictionary<string, string>();
dictionary.Add("name", currentUser.Title);
dictionary.Add("email", currentUser.Email);
dictionary.Add("timestamp", ts);
}
With this code, I am also able to grab content from the 'web' variable.