My first fixed gear was a road conversion built on an old Giant lugged steel frame. Bought a Bianchi Pista used because I thought it was more legit and it was a good deal. I was super excited for a "real" fixed gear. Ride quality was pretty meh and never really like the bike much and ultimately ended up selling it and going through another few bikes before settling on a Soma Rush. Soma has similar geo to the Bianchi but better steel and a much improved ride and feel.
The Pista is a very entry level frame priced like something fancier.
I wouldn't spend too much money on improvements for the Bianchi or if you do keep an eye on things that are easily moved to another bike when the time comes (bars, stem, wheels) rather than things that are more bike specific (headset, bottom bracket etc) that might require a different size for a new frame.