I understand that using a soft iron core or a ferrite core for an electromagnet increases the strength compared to an air-core. I'm going to be using DC current and want the core to be 5-8mm in diameter and roughly an inch in length. So where would I be able to find a soft iron core? I found this on Digikey and not sure if it's the right material.
For a DC electromagnet unless remanence (keeping a field after you turn it off) is a problem, you just want a high permeability so you can get a high field strength. If you check out the Wikipedia article on magnetic permeability there's a nice chart that I won't bother copying to here.
From the list you can see that you obviously need Metglas because it has the highest permeability! Nah, sounds expensive... 99.95% pure iron annealed in hydrogen? Nope... Permalloy, NANOPERM® (The trademark symbol means it's cheap, right?)... Then we get down to cobalt iron and iron. Probably iron is your best bet for reasonable cost, although if your core shape is a rod of reasonable length you could experiment with cutting the fluted part off a cobalt steel drill bit without spending too much.
Think about the shape of your magnet and how you want to use it. The magnetic flux has to make complete loops, so in a rod type core for example, the air around and outside the magnet is part of the magnetic circuit, where the air part of a U shaped magnet is just the space between the ends of the U.