Shane...
Sounds like you're getting into bullseye pistol. Ultimately you'll need to do 3-4 things to your pistol to make it shoot better than you can hold.
You will need a match barrel. Kart or Barsto make some mighty fine match barrels, with Kart being just a bit less expensive. These will have to be fitted by a qualified gunsmith. You'll never shoot better than the inherent accuracy of a properly fitted match barrel.
You'll need a match trigger and trigger job. Videki makes a good match trigger with triggerstop at reasonale cost, plus they are available in long and standard length, depending on your taskes and length of your trigger finger. The trigger job is another that a gunsmith should do for you. This smooths up the parts and sets the trigger weight to 3.5 lbs for a wad gun, or 4 lbs for a hardball gun. Done incorrectly, the pistol can be unsafe and be prone to double or even worse, go full auto. Without a quality trigger, you have a difficult time executing the shot with any accuracy.
You'll need match adjustable sights, and Bomar is the universal standard, if you are going for open sights. If you opt for a red dot sight, Ultradot is not a bad choice. You need a clear, precise aiming reference that match sights privide.
Your Springfield is one of the best platforms to build up a full match grade pistiol, plus you don't have to do it all at once. You can space these modifications out over time as finances permit. It is a toss up in what order to do the mods, but I generally suggest a trigger job, match sights, and finally a match barrel. Some standard pistols have adequate barrels from the get go, so that is why you can wait a bit. In your case, the first option may be sights.
If you are not getting into bullseye in earnest, then less expensive sights will do. You may decide that a set of high visibility fixed sights meet your needs. MGW makes a nice set.