Most laptops can only output from one display-type port at a time, so HDMI OR VGA in your case, usually the second port is only provided to improve compatibility.
If you really want the dual screen set up, you can go for a splitter, although from what I've heard these can be funky and come with their own problems.
The other option is to get a USB display adaptor, ideally USB 3 if your laptop supports it. If you have to use USB 2 then you may notice a slight bit of lag on the display, it will be fine for browsing email and facebook, but movies and games won't look smooth. USB 3 may have much greater performance here.
That said there are plenty of laptops out there that do allow more than one external screen natively, so my advice would be to buy the new monitor and if it doesn't work then buy the USB adaptor.
As an aside VGA doesn't have poorer picture quality than HDMI, I'm using dual monitors now. Two identical screens, one through VGA one through HDMI and no noticeable difference in picture quality (besides the contrast on the HDMI which I've been meaning to fix for weeks). The issue with VGA is that being analogue there is more room for interference and effects of bad/long cables to degrade the signal. If you do go for VGA I'd advise choosing a cable wisely, no need to go super pricey but avoid no-name cables and stupidly low prices. That said VGA is dying rapidly so try to get a monitor that does VGA and something else like DVI if you can.