Upgrading my system

well, if you have to change RAM anyway, then one more stick won't break the bank.

Now, the PSU should be able to output the watts it's rated for, Liteon is a relatively reputable manufacturer and the sticker shows that it's a PSU made specifically for Lenovo, that will likely have looked that it wasn't crappy as that's bad PR.

At the moment your graphics is the one integrated in the processor (physically inside the processor chip). It sucks from all points of view, but if you don't need to play games nor watch movies it's ok and it needs little power.
The slot where you could put a decent graphic card is the long black one, it is between the three small black ones and the CPU fan. It's likely a PCIe gen 1 so it's not stellar, but will handle most graphic cards below 250$ just fine.

The three small black slots below that are PCIe 1x slots, and can run various kinds of other expansion cards, like for RAID, Wireless, TV cards, more USB or Esata ports and more.

If you want to get a decent graphic card in there (better than the HD 6450 anyway), a very good choice would be to buy a reliable PSU with a bit more watts of output than the one you have already. By reliable I mean ones with a well-known brand, that have the 80plus certification and sticker as I linked above (that means someone tested it to make sure it does not meltdown if asked to run at 100% capacity). They will cost 50$ or more depending on wattage, if your wallet isn't hurt it's very likely a scam. Remember to use the graphic card hierarchy chart and the PSU calculator to make sure you are getting the most bang for your bucks AND that you have enough power to run everything.

You can also upgrade your processor (although this will break your current budget I'm listing it anyway).
As this pdf from lenovo lists what processors they sell the same rig with in the page about your model (page 5).
For the sake of giving a rough idea of the difference in processing power I'm using CPUbenchmark's points as that is a trusty site I use as well.
The processor you have now is the pentium G620 that has 2300 points.
You can install an i5 2310/2320, at 5827 and 5760 points respectively. Here is one on newegg at around 200$
You can install an i7 2600 at 8421 points. Here is one on newegg at around 300$.
You can install some i3 2100 and 2120 too but they aren't so much more powerful than your processor to be worth spending any money on (if you found some for free or for very cheap it's another matter).
I personally like Ebay and buying used processors, as you pay a lot less, and as long as they aren't "k" ones (the overclockable versions) the chances they don't work is slim.

All in all, not that bad. I thought it would be much more locked down. Both the i5 and i7 it can take are good processors.
If you tell some more details about what you want to do with the computer I can give better directions, as I usually assume it's for gaming, but you may have other needs.

Nice cases like you ask cost around 50$ (some much more), and more fans are not necessary if you don't overclock (and you cannot overclock with that board)
Given your limited budget, I'd recommend focusing more on power than on looks.

Anyway 300$ you say? Then no ram, a good 450W PSU, and a the best card you can get with what is left (in this order). And leave for the future the upgrade to the i5 2310 as that's another 200$. Consider that if you sue an additional graphic card, the onboard one is disabled and should leave more RAM for you, so even 4 GB isn't that bad.

Still, I'm very very tempted to recommend you to ditch both current processor and motherboard, and take an asrock extreme4-m plus a Win7 Home Premium OEM licence with a i5 3570k. Processor sells for 220$, mobo for 140$ the new Win7 licence (because the windows installed on your current machine needs a new licence if you want to migrate it to another machine) for 100$. Since you have one working 4GB ram stick, if you move it to the new board you can use it for a while and save a bit. It's not that bad.

The i5 3570k is currently one of the best if not the best processor for gaming (surely the best for the price, and performs better than the best processor you could mount in your current PC even if not overclocked), and it has pretty good integrated graphics, about on par with the HD 6450 card (so it can play HD movies fine, it can go up to a resolution of 2560x1600, which is a 30"-ish monitor or TV).
Of course the HD 4000 graphics is not going to play very demanding modern games like say BF3 (while the processor could run it easily if you installed a graphic card).
If you are accustomed to what a G620 can do, that's a huge leap forward.

The board isn't crappy either with 2 last-generation PCIe 16x slots where you can mount and use two modern graphic cards if you want. Plus you get the usual stuff like native USB 3.0 ports, a e-sata port, 4 SATA III ports in case you want to buy an SSD, HDMI port to connect it to a TV, and a PCIe slot in case you want addon cards.

This is 460$, right, but it's the foundation of a good modern rig with significant horsepower even as-is, with still plenty of room for real upgrades and overclocking in case you need it, that will run fine in your current case and with your current PSU (not overclocked).
The real elephant in the room is the 100$ Win7 licence though. After you tried transferring the HDD with Win7 on it hoping it just accepts the same licence, you can try the phone activation, you tell that the older board died or whatever, and hope the guy at the phone gives a new licence. It works a lot of times.

Next recommended upgrade would be a 50$ for a good 450W psu and some 100-200$ for a good graphic card plus a better CPU fan in case you want more and want to do some overclocking.

Yeah, I'm a salesman , I tried to give you the whole picture, probably someone will start bashing me for my choice of motherboard, and maybe offer something better, so keep an eye on this thread. But frankly, at most you are going to save 60$ even if you drop every feature saved for one PCIe graphic card slot.

Took a loot just at the reviews on NewEgg, and they seem decent.

Reviews on newegg and any other site selling things have to be take with a grain of salt, as the people posting there are just common people without any real experience in the field. If you want to know what actual experts think, use google, search for the device name and "review", you'll find sites like tom's hardware, anandtech, 3dguru, and others that reviewed the product and give their expert opinion on it.

I would like another screen, since I always have a lot of projects going on at once.

both options will allow you to use more than one screen (the one in your budget and the second one that is well over 150$ out of that), as pretty much all cards have 2 or 3 outputs and can handle two-three screens. Gaming on multiscreen requires a very powerful GPU, but when you aren't playing games pretty much all cards can deal with multiple screens.

I'd like to get another, doesn't necessarily have to be the same size, but hopefully somewhere close.

Do you have a TV with a HDMI port? You can use that as well, as long as the graphic card you buy has an HDMI port (most do). Of course you need a HDMI cable, but that's a lot more space for your stuff.
As for buying a good screen, I can only give you tips on what to look for as screen models change faster than printers and it's hard to find a specific model after a few months you first saw it. Try to buy from a shop where you can see it working and play with its color/contrast/whatever options and look at it from various angles to check if colours remain correct if you do so.

Also, is the Windows 7 OEM License a worthy upgrade over a monitor?

You need the licence to run Win7 on a new motherboard (as it's linked to motherboards), as the licence you have is linked to your current one.
But as I said, even if you go the conservative route and just buy ram, a psu and a graphic card you can use two or three screens (again, all for work, only one when gaming).

RAM shouldn't break the bank, any sticks you think are worth your money? Let me know your thoughts.

Good brands, like Corsair, Kingston, IGSKILL, Crucial, and probably some other I'm forgetting.
They all have three general RAM lines, normal, with low profile heatsink, with big heatsink.
THis page from Crucial is an example.

What you want is the ram without any kind of heatsink, where you can see the naked board and the chips, which in the example is the ballistix sport.

Heatsinks are supposed to help you overclock the RAM, but as that's totally pointless even for serious gamers with current processors, it makes no sense to waste money on useless features.