Yesterday, I rode my bike for the first time in months (30km commute one way), and on the way home, my chain dropped off, and I tried to pedal forward to pull it back on, but ended up twisting it pretty badly. I found a shop half way home that replaced the chain in about 15 minutes, so that was good.
The unfortunate part was that I planning on changing derailleur and cassette (28t to 32t), so now I have a chain that is cut too short. Now, I'm looking to order thew new parts, but need some advice on what I'll need to lengthen the chain. Will I need a multiple connector pins/master links? Any good advice to go about doing it? It will also be my first time change a derailleur so a little nervous about that too.

You can extend the chain length with master links and maybe a section of chain. In the second half of this video he does some chain lengthening:

I want to go from an 11-28 to an 11-32, when I go on long rides in the hills, I run out of gears... I was actually planning on doing the switch soon, but ran into the chain problem last night. I have all the links that came with the chain, and am really wondering if I'll need two masterlinks or connector pins to lengthen.

The other option is paying a premium and going to shop, and having the cables done at the same time too.

Do not drive the old pins back in. Use a master link. The new narrow chains have pins that are peened after being driven in at manufacture so they are now a little mushroom shaped. They enlarge the hole when you drive them out to make changes. If you drive them back in, it is now a loose fit. Odds for one of the pin ends slipping and the link coming apart are quite good.

I've lengthened chains by simply re-attaching the links with a second master link. So you need two masterlinks: the one that's currently connecting the chain, and a second one to attach the links you need.

Thanks everyone. I ordered two KMC masterlinks (was a set), an 11-32 cassette, and a derailleur to handle the larger cassette. Hopefully, I get through everything without too much swearing or skinned knuckles. Looking forward to spring and huffing and puffing up some climbs as better riders drop me.

As for installing and adjusting a new derailleur. I find it very simple - easier (to me) than installing a front derailleur. I recommend using the derailleur's instructions for this, maybe a video to make it clearer if it isn't. The diagrams are usually very clear. There's plenty of videos online. Bolting it on is simple. Then you just adjust the high and low limits, which are very simple and objective settings. Then the cable and indexing. All of this - as well as chain length - are pretty simple and will be in the instructions and many online videos and other sources. If you have any specific questions for the installation and adjustment, just post a new thread here.

Other than possibly a chain tool (only needed if your spare links are too long), there's really no special tools needed. Hex key (aka Allen wrench) multi tool and regular, small screw driver if you don't already have those.

Does my chain looks too short? When I sized I cut where I cut where I couldn't add the master link, so maybe one link short? It does shift into the big gear... though I'm currently trying to adjust my front derailleur.
One more link? Enough of a bend?

If the derailleur is free to move even further, and it changes to the largest gears without a problem, I would not lengthen the chain.

^Thanks
Shifting to the big ring didn't seem to be a problem. However, I had trouble with chain rub on the front derailleur, and totally ****ed the front derailleur and frayed a brand new cable trying to fix it... so frustrated.