You could start by trying to solder to something less populated like this little daughter board:
View attachment 39952
This should be an easier starting point since these aren't like the really small surface mount components on the other boards. If you're having trouble getting a shiny, smooth, round, joint, then make sure you're using a decent quality flux. Rosin core solder doesn't really count as using flux since the rosin isn't as good as regular flux, and adding more rosin always implies adding more solder.
I recently made a guide for micro soldering to the Wii that you can find here. While it may be focused on soldering to the Wii, it also goes into detail regarding what flux and other soldering equipment I use. I'm bringing this up because half of the struggles people have are related to their low quality equipment. Remember that low quality equipment will leave you with low quality solder joints!
While this is a resourceful way of teaching yourself to solder, you may be better off purchasing a soldering kit from Amazon if you're having trouble even with decent quality soldering equipment.
Finally, you seem to be concerned with safety (rightfully so!) so here's what I can tell you based off my (admittedly limited) knowledge:
First off, never solder to a powered circuit. If you're making a portable, this means to never solder to the board while both positive and negative sides of the battery are connected to the console. In the case of your old DVD player, make sure you have it unplugged, and also disconnect boards like the power management board from it since, that board uses higher voltage components.
Second, avoid touching capacitors and transformers like what I've circled here with your bare hands:
View attachment 39953
View attachment 39954
The smaller ones are probably less of a concern, but just try not to touch them with your bare hands (or anything that isn't insulated between your skin and whatever you touch it with). Most of these are through-hole caps, so if you want to discharge them you'll probably need to go to the other end of the board where they're soldered. Regardless, it is best to avoid touching them altogether.
Also, I probably don't have to say this, but don't ever connect positive and negative lines to each other. Doing so can destroy your circuit at best, and cause fires or (in the case of batteries) explosions if done wrongly enough. I use a multi meter to verify that any and all voltages aren't shorted with each other and or ground before I test anything after I solder, and you should too.