I want to record some covers into my laptop, but I don't understand the technology part yet.
First off:
I'm using a Schecter C-1 Lady Luck: https://www.ultimate-guitar.com/reviews/electric_guitars/schecter/c-1_lady_luck/index.html
And a Line 6 Spider IV 75w: https://www.ultimate-guitar.com/reviews/guitar_amplifiers/line_6/spider_iv_75/index.html

I'm not looking to record with a mic and this is where I get confused. Do I need a preamp or a DI-Box? If I select a channel on my amp, will the sound be the same going through the box? How much does my sound card affect the sound quality?

I'm a rookie at this, please understand if something seems very dumb.
Bombard me with questions if you need to get further info.
You'll want an interface. The soundcard on a computer is not designed for high quality recording. As well, the connections on the soundcard are 100% not optimal either, as they generally use 3/8" TRS jacks whereas all mics use XLR.

Get a 1 or 2 channel USB interface. They usually come with a lightweight DAW program as well, which will suffice.
Quote by Watterboy
Do you have any dilithium crystals or fresh warm dumps for sale
Pick out a nice interface, find some great software based amp sims and effects, plug in direct and wail away. The Spider can be used as a platform to hold your interface or to lean your guitar against while taking a break. Just don't plug it in while recording or bad things will happen.
"Your sound is in your hands as much as anything. It's the way you pick, and the way you hold the guitar, more than it is the amp or the guitar you use." -- Stevie Ray Vaughan

"Anybody can play. The note is only 20 percent. The attitude of the motherfucker who plays it is 80 percent." -- Miles Davis

Guthrie on tone: https://www.youtube.com/watch?v=zmohdG9lLqY
You can get an interface and use sims in the box by going direct (either freeware like LePou or paid one like Ampltibe, etc.) or get an interface and mic or use direct out of your amp for the sounds. You'll also need some kind of DAW to make this work - Mixcraft, Tracktion, Reaper, Studio One, Pro Tools, Cubase, Sonar, etc. You can get a computer interface with an instrument input and mic input / XLR just in case you decide to experiment with vocals later or mic your amp. I prefer to have as many options as possible so don't want to be locked in in one way of doing things. Most new interfaces on the market will have one or two inputs, some have combi 1/4 and XLR inputs. Look up what is available and once you narrow down the options hit us back with what you're buying so we can advise you further. A trip to Guitar Center or similar won't be a bad idea just for an informative visit.
Quote by Random3
It will work, yes. However, the 2i2 doesn't have a pad, which means you will likely get signal clipping with medium to high output pickups. Spend the extra £30 and get a 2i4.

There is a volume control on the guitar - so why would you need a pad for recording it?
If you're just recording guitar, also consider using the JamUp Pro app for iOS. It is fantastic. I use it both live and in the studio.

You'll need something like an iRig (recommended = iRig HD) to plug your guitar into your iPad/IPhone/iWhatever, and the JamUp Pro app is about $30, which includes all the amp and effects models.

You should be able to go directly into your soundcard, but it won't sound as good as if you were to get an interface and run the headphone out from the iDevice into that.

Could I get some more talent in the monitors, please?

I know it sounds crazy, but try to learn to inhale your voice. www.thebelcantotechnique.com

Chris is the king of relating music things to other objects in real life.
EestlaneAlo your setup works, supposedly the clipping issue on earlier Focusrite 2i2s has been fixed, so if you buy with return policy this might be the ticket.
For similar $ also look at:
PreSonus, Roland, Steinberg