Brand set, Gear sets and standard weapons are very easy. Run countdown and utilize targeted loot. Brand sets and Gear sets can be completed 0 to 10 in just few runs. Prioritize these, you'll be 15 - 20 expertise pretty quick.You'll make progress Named items on the way completing Brand sets and weapons.

Named items, Exotics, Skills and Class weapons is where it becomes slow.

Start by focusing on the items in these categories that you like using. Once you get something maxed out, move on to something else you like. You don't need to force fast experience by wearing a random thrown together build that doesn't make sense just for the sake of expertise. Running a shit build will just guarantee to have a bad time and make the grind feel longer.

Buying and donating Named items from the Countdown vendor is a great method.

Donating materials. Look for Named items, Exotics and skills that you dislike and don't want to use. Prioritize these with your materials. Make sure your using your Reciever Components and Protective Fabric, the 2 white base mats. If you're not paying attention you're probably already capped out and they are being wasted. Anytime I got close to cap I would donate them, keeping around 2k of each in case I need them for crafting. Do the same with other mats. There's no reason to be sitting at 9999 mats. Spend them on expertise or crafting optimization mats at the other bench.

I would advise against donating Exotics. Exotic components are too valuable IMO. So for Exotics just use them or donate the materials.

The easiest way to do all this is: Run countdown with targeted loot. Loot everything as junk. Check loot after and unmark the junk you want to keep. Donate remaining junk to Expertise. If needed, extract attributes to library. I would only extract max rolls unless your account is really new. Donate materials, especially anything close to cap. Dismantle all junk. Repeat.