Hello all. This is my first post here, but have been browsing the site for some time and find that the advice offered is very helpful. Thought I'd reach out to get some advice of my own.

I've got a 2012 Specialized Sirrus Sport. When I purchased it, I was only biking a measly 1 mile a day round trip to and from work. However I've since moved and am doing 10 - 11 miles per day round trip. The bike is great, but I've been considering swapping out parts here and there to make the ride more... fluid. For example, I picked up a nice set of Nitto Grand Randoneur dropbars which I'll be installing soon, once I find the answer to my question.

I'm trying to make the bike practical, comfortable, and suited for lengthier bike rides. I like the different options of drop bars, more than the straight bar the bike came with.

I've been researching the different types of shifters for the new dropbar. It seems there are bar-end shifters, and STI shifters. I think I'm going to go with STI shifters because 1) I like the way they look and 2) I don't think I'd feel super confident having to reach to the end of the bar to shift. I'm looking at Shimano Sora 3x9's or some in the Tiagra line. But I can't determine if they'd be compatible with my Shimano Acera derailleurs.

I did come across a PDF from Shimano showing parts compatibility but nowhere did it mention the Acera's. My assumption is that I'll need to pick up some new derailleurs too, but was hoping to hear from you all first.

The derailleurs I've current got installed:
Shimano Acera FD-M360
Shimano Acera RD-M390 GS

And the shifter set I'm most interested in is:
Shimano Sora ST-3400

Are the two compatible? Are there better recommendations (keeping budget in mind)?

Thanks in advance,
Chris