"SPS" is a model designator, used by Remington. A model 700 SPS, is the cheapest version of their model 700. It has their matte "special purpose" finish, and is in a BDL configuration.
Remington used both of these as model designators. The ADL's were the cheaper versions, and the BDL's were fancier versions of the model 700. Today, "ADL" and "BDL" are commonly used as slang to distinguish the different magazine configurations. An "ADL" has a blind magazine that must be both loaded and unloaded from the top. A "BDL" is loaded from the top, but has a hinged metal floorplate to allow faster unloading of the magazine from the bottom of the rifle.
Talley is a brand name. They make a couple of different varieties of scope mounts/rings. My first choice for hunting rifles is their 1-pc Lightweight series. They are very light and really strong.
Link
Tuning a trigger usually involves lightening the pull weight, and taking all the slack or "creep" out of it. A tuned trigger should break cleanly. Personal preference determines how light it should be, but somewhere around 2.5 - 3.5 pounds of pull should be decent for a hunting rilfe.
AI is short for "Ackley Improved". More information can be found here:
P.O. Ackley
I got the Remmy action (700), but what brand barrel?
There are lots of very good aftermarket barrel makers......Lilja, Kregier, Hart, Pac-Nor, Shilen, Bartlein, Douglas, etc. You can Google each manufacturer to get more specifics and prices.
The Remington factory triggers are usually pretty heavy (the new ones are a little better), but can be adjusted fairly well. A "tuned" factory trigger is a decent set-up, but there are several aftermarket triggers as well......Rifle Basix, Timney, Jewel, Shilen, etc.