Ok, I'm gonna go a bit into detail here...
Unfortunately you'll need a bunch of drives to cover all those consoles properly, that is if you really only want to use your PC and not the modded consoles to extract the discs (which in many cases saves you quite a lot of time/headache).
PS2 is the easiest of the bunch. As you already found out, these are plain DVDs; any DVD drive will read these discs just fine.
As for Gamecube and Wii discs, you're correct, you need the LG GDR-8161B, GDR-8162B, GDR-8163B or GDR-8164B and a program named FriiDump. Extracting the GC/Wii discs is pretty slow with these drives (you can expect about an hour per GC disc, several hours especially for dual layer Wii discs), so a modded Wii with CleanRip is a more convenient solution.
Xbox and Xbox 360 are quite similar to GC/Wii in that you need a "special" drive, namely a Kreon compatible drive. Kreon is an alternative firmware that allows these select few drives to see the game data on the disc. You can dump the discs with Xbox Backup Creator pretty easily then. If you insert an Xbox or Xbox 360 game disc into a non-Kreon DVD drive, you'll only get a DVD-V partition containing a video telling you that this disc is meant for playback in an Xbox or Xbox 360 console. An example for such a Kreon compatible drive is the Samsung SH-D162C; I can't really remember any other drives, so you'd have to google that.
PS3 and PS4 can be read with most if not all standard Bluray drives.
Xbox One discs seem to be readable by certain drives as well, but I have not looked into that topic at all, so I can't give you any reliable information here.
Wii U discs are impossible to dump on a PC so far.
With CD-based consoles, it really depends whether the disc contains audio tracks. If it doesn't, you can use any CD or DVD drive to dump the disc (for example, 3DO discs don't contain audio tracks, PSX discs sometimes do). If the disc DOES contain audio tracks (as is the case with Saturn discs, Sega CD discs, PCE-CD discs, ...), you'll have to decide... if you don't care about slight (that is, completely unnoticable) inaccuracies, you can use pretty much any CD or DVD drive for dumping them. However, if you're like me and you'd prefer to have super accurate dumps which you can verify against databases such as redump etc, you're gonna need a drive that supports overreading into leadin/-out. I highly recommend a Plextor drive such as the PX760A -- these are excellent drives with great error correction.
As for GD-ROMs (the discs used in the Sega Dreamcast), I highly recommend not dumping them via PC. These discs consist of several "areas"; the one that contains the game data is written in higher density, so it's not accessible on a standard drive. Technically, it's possible to use certain drives -- such as the aforementioned Samsung SH-D162C -- and trick them into reading the discs. The process to do so is really really complicated and involves taking your drive apart. If you're interested in the gruesome details, see http://forum.redump.org/topic/2620/dreamcastnaomi-gdrom-dumping-instructions/
TL;DR: If you really want to do everything via PC, I suggest:
-> LG GDR-8164B (or (8161-8163, whichever you can get) for GC/Wii
-> Samsung SH-D162C for Xbox/Xbox 360 and DC (if you're brave enough...)
-> Plextor PX-760A for CD-based media, i.e. Saturn, PSX, Sega CD etc.
-> any Bluray drive for PS3/PS4
PS2 would be possible on ANY of these drives. CD-based media too, so if you don't need super accurate dumps, you don't need the Plextor.