I know Brad
Also you need to get to know brad really well since that is the exact type of nail to use for shoe molding.
What is the difference between a Finishing nail and a Brad nail
Well the nails are different gauges with the brad nail being thinner. And also brad nails usually have smaller heads. But really the best way to describe a brad nail vs a finishing nail is a brad nail is not made to be hammered into anything by a human - maybe Zeus on a good day. Brad nails must shot into the material on one go as they will deform with any sort of non-straight pressure. We have messed around on sites to see who can get a brad nail the furthest into a piece of scrap wood... if you made it half way without bending you win!
A finishing nail therefore is something that can be hammered. Finishing nails can be shot from a gun but it will require a bigger gun and quite frankly you don't do this much as the point of using a brad nail is to create less of a hole that has to be touched up. Using a finishing nail often defeats the purpose (there are times when it may be required but quite few). Also finishing nails for the most part are round, while brad nails are usually a rounded rectangle.
Can I hammer in my finishing nail?
If you don't know the answer then the answer is no. Hammering in finishing nails and using a punch - shown below - is a skill. First you have to get the finishing nail hammered in the right direction, precisely. Then once you get close to even with the wood you have to use a punch to embed it.
On a carpentry skill range this is a solid 8 out of 10 in difficulty and unless you feel you are already a pretty skilled carpenter this is not a skill you should practice developing.
Installing shoe molding now... well that is a 10 out of 10 on the difficulty range hammering finishing nails. Even using a nail gun with finishing nails is extreme for shoe molding and you will inevitably split the molding.
I usually set shoe molding with a bead of caulk, give it some time and finish with 1.5" brad nails.
What do you need to buy?
Given that space is not an issue every homeowner that is going to venture into their own repairs is best to buy a nail/stapler combo kit and a compressor. There are just so many uses that it pays for itself quickly and saves you loads of time.
Here is an example... not a recommendation at all. The basics that any home owner needs is a framing nailer, a brad nailer and a stapler. As long as your kit includes those that is a good start. Some kits may include more than one size stapler, some may include air attachments, and some may include two sizes of brad nailers.
This kit was $230. You can get other kits like this with more or less things in the $180-300 range. You get a 5+ gallon compressor for ~$100 and you are good to go. You can go electric but these have more problems and are way more costly individually. Also the compressor has a TON of uses around the house from blowing up tires to painting to floor jacks.
Framing nailer = framing walls, building big things
Brad nailer = trim/molding
Stapler = repairing furniture, securing thin wood panels, crafts
Isherwood's answer does a good job adding the benefits to the air compressor and pneumatic nailers. I would add though that the biggest benefit is that you will be able to adjust your pressure better with these. For example for your shoe molding, your pressure needs to be just right. Too much and you have a divot. Too little and you have 1/4" sticking out and those things are hard to punch in.