I have a flower!

Newbie gardener here (1st house, 1st garden) live in zone 4a and we've had a really crappy long cold spring and early summer that (some say)shortened the growing season, but things seem to be coming along fine.
I have a question about my strawberry plant. I bought one from Home Depot in early May and kept it indoors, transfering it to a MUCH bigger pot because I know they need room to grow. It was marked 'sequoia strawberry' and I figured that was the variety of strawberry. Research online indicated that it would probably bear no fruit its first season and that I should keep it in the pot and plant it in a raised bed next year (2009).
I watered it according to what the general consensus of websites was (keep it damp but not running with water, add fertilizer every once in a while) and watched it flower. One flower. Eventually in late May it produced one strawberry, which when I ate it was pretty good, all things considered. The plant was pretty stringy and the strawberry was actually weighing down the plant, but it still looked healthy, so I didn't worry.
After I ate the strawberry the plant seemed to go into dormant mode, or something. I kept watering it, and soon after moved it outside (still in its pot) to our new yard. The amount of rain we've been getting seems to agree with it -- now it's putting forth new leaves like crazy which are coming up in clumps, very low to the soil (more like how I've always seem strawberry plants look) and the leaves themselves are a very healthy color and texture. Very soon I'll have to transfer it to another, even bigger pot (prepatory to bringing it in for the winter and then actually *planting* it next year).
And I have another flower! With one more about to blossom. This surprised me because I figured that my plant was done for the season.
My question is this: Can I expect more strawberries out of this plant? Should I let the flowers turn into berries? (Some of the websites seemed to agree that I ought to pinch the flowers off, but my plant looks pretty healthy to me with them on.) Advice from people who've gardened before would be good.
The Newbie Gardener
Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
Add image file

HomeOwnersHub.com is a website for homeowners and building and maintenance pros. It is not affiliated with any of the manufacturers or service providers discussed here. All logos and trade names are the property of their respective owners.