I think this is a really interesting start. I seems very much like a ordinary crush-on-a-teacher situation (who hasn't been through that embarrasing moment?!) However the reactions of the older generation intrigue me and I wonder exactly what is going to happen! There is of course the complication of the friendships between the adults involved.
A few small things: I find it can be helpful to put in some sort of break before the text of the story itself, just to stop the story notes running into the start of the story - visually I personally just find it neater and easier to read.
I also found it odd that they called the two recent wars the first and second wizarding war. There was at least one war before them and I can't believe they're haven't been other wars. I guess I would put an additonal descriptive word in there, even if it just the First/Second Great Wizarding War, to distingush them. I am interested to see how Lily learning about her parents/relatives actions in the war affect her and her crush. There is the potential for a fair amount of conflict and emotional distress given how fresh and recent the war will be for those who were involved in it.
The characters feel reasonably well fleshed out, particularly Lily obviouly! I think the first person will work well as I can see much of the conflict/plot progression happening around Lily's thoughts and feelings rather than a lot of action. I liked Lily's musing on Neville and his hands ... it worked really well as a kind of 'oops, that went further than I thought!' moment for Lily. ;)
Look forward to seeing more!