Radeon R7 240 and AMD Dual Graphics functionality on an OEM motherboard

My recommendation is to forget the dual graphics entirely. It's bad. Real bad. It'll pretty much make your purchase of any graphics card, to put it kindly, useless. (It's actually more of a downgrade than an upgrade.)

Just use the graphics card as the graphics card, and the APU as the CPU. Don't try the Dual Graphics nonsense.

It depends on how much you're willing to spend. There is the 750ti, 1050, 1050ti, although the latter two may require more juice than the 300W power supply of unknown make and quality can provide. There are also plenty of older options that are still faster than an R7 240, like the old HD7750 and 1gb ddr5 gt730, if I remember correctly. The Sh*t Card Showdown series by Socket Sanctuary explores plenty of these older options that are still viable cards today.

I once had an R7 240. Bought the thing for $30+tax, and overclocked it as far as it would go (bad idea, don't do this). I later scored an HD7950 (r7 280, r9 380~ish) for $100 on newegg, the last one in stock in fact, and realised how poor value for money the R7 240 was. I recently gave away my r7 240 to a friend who needed a back up card when his was in transit during an RMA. If I still had it, I probably would have sent it to you for free, provided you paid the shipping cost.

I fail to find any detailed information on the power supply that's in these Gateways, but if it following the standard ATX format, I'd recommend upgrading it to a better/more power unit, opening options for much better graphics cards and future upgrades. If anything, it'll be one of few components that can be transferred from PC to PC. For an extreme budget, if you can find an Antec VP450 for around $35-40, it's a sturdy little ah heck that's closer to a typical 500W supply than its rated 450W (overbuilt and underrated). I personally vouch for the sturdiness of the little ah heck. I pulled 550W out of one for a whole year, and it's still going solid. For more juice at a bit more cost, there's the Seasonic S12II 520W, or 620W if you so desire. For more information, the PSU tier list here on the forums is a great resource. If you want super technical details, the jonnyguru site is a bit of a hassle to navigate, but the PSU tier list is largely based on jonnyguru's information.

TL;DR I'd put in a better PSU, which will open up a variety of options for graphics card upgrades. And maybe even a CPU upgrade down the line, if you so desire.