I have been mainly working with "real" sound effects but in certain games I play I have no idea how they get certain sounds (by ear, it seems to be synthesized to me). The YouTube video linked has two example sounds, a power-up sound from The Binding of Isaac, and a collect sound from Spelunky. Any tips on software, plugins, places to start, etc. would be helpful.
Good start for retroish synth sfx is basic square wave with invidual ADSR for both pitch and volume. Experiment with different kind of parameters and think of the direction where sound is going. Add some effects and filtering if you need to make it sound more modern than 8-bit computers and consoles.
As for the SFX heard in the clip.
First one sounds like filtered explosion sound layered with synth playing very short notes going up maybe half octave per note. Synth patch sounds like it could be comb filtered retro arpeggio sound, but I might be totally wrong :) There is a delay effect in synth which has hipass filtered feedback.
First part of the second sound could be from anything. Very distorted and bandpass filtered simple synth sound with very fast decay could be a good starting point. Second part could be resonating bandpass filtered square wave with rising pitch and decaying volume.
I'm pretty sure I'm not very correct with this analysis, but at least it should give you a good idea of what kind of things you can try :)
Personally I would start with a sample that you like, maybe something that sounds like the beep. I'm a linux user, so I'd open up Ardour 3 (which is also on Macs, and is totally awesome) and then for each layer in the sound, have a new track.
I'd put a delay on the beep, so it sounds like it's decaying for the first sample. I'd add an explosion sample, low pass it and then window it. Then add a few other bits and pieces in to fill out the sound. I've had great success using softsynths like ZynAddSubFX, which, although the learning curve is steep, you can create anything you like.
You can either build it up with wave forms and place them where you want to, or you can actually sequence them together using midi, being driven by a softsynth. The sample is also heavily compressed.
This is not the be all and the end all, it's just a pointer to where to start. Break up the sound you want, and add more tracks until you get it. If Ardour scares you, start with Audacity and get all the LADSPA/DSSI plugins.
Basically start somewhere. You don't build a building without stages and sections.