Hi and welcome to POTN...
Odd that camera coming with two teles and no shorter lens. Usually in kits they have more complementary lenses.
The 18-55 IS is a decent lens optically, but very "entry level" and inexpensive. Similar in build and function to the EF-S 55-250, except for the focal lengths of course. The two are often offered together in kit, but usually with the more entry level Rebel series cameas, not with the 60D. An 18-55 IS would likely be your cheapest option (just be sure it's the "IS" or the latest non-IS Mark III version, for best image quality... you don't really need IS on an 18-55 lens, but what the heck if it doesn't add to the price).
Usually the 60D is sold in kit with the EF-S 18-135 IS. That's a relatively new, nice walk-around lens, covering a broad range, but it is a bit more expensive. In kit it usually adds about $300-350 to the price of the camera. Bought separately it's about $450. You might find used for less, perhaps around $400.
The 70-300mm is an EF lens, so is usable on both crop and full frame cameras. The above are EF-S, so only usable on crop cameras (such as the 60D). This might never matter to you, if you never "upgrade" to a full frame camera. But it might make the lens a little more valuable. This is also a USM lens, so it's focus should be faster, quieter and more accurate. I haven't used it so can't really compare with the 55-250 optically. If it were me, I'd keep the 70-300mm. (There are a bunch of 70-300 or 75-300 lenses, which vary a lot in quality... if you go looking for more info about it, be sure you are looking at the right lens.)
The least expensive, quality "walk-around" lens is probably the EF 28-135 IS USM that was sold in kit a lot with 40D and 50D. It's still sometimes offered with 7D and 60D. There are tons of them around, so they are easy to find pretty cheap... There are nearly always a couple on my local Craigslist and recent asking prices have been about $200. This is an older design EF and USM lens (the newer 18-135 is not USM and is EF-S). It's got the best build of any "kit lens", except for the 24-105L that's offered with the 5D Mark II. It's quite good optically, at all but the longest 135mm setting where it gets a slightly soft if used wide open. I like the focal length range as a convenient walk-around, except it's not wide at all. If you want wide angle, you'd need to add another lens for that purpose, now or in the future ( for example Canon 10-22, Tokina 12-24, Sigma 10-20, Tamron 10-24). That might be a good thing... since you'd have a very complete and extensive range of focal lengths in three lenses.
A lot of people buy the 50/1.8 as sort of an introduction to prime lenses. If you are an old 35mm film shooter, just be aware that a 50mm lens is not a "standard" lens on modern crop sensor cameras. It acts as a short telephoto lens on those cameras. It's nice for portraits, for example. The 50/1.8 is cheap but capable. I wouldn't buy it if planning to use it rather rigorously... It's not all that good build. And it doesn't have USM, so is a bit slower, noisier and less accurate focusing. The Canon 50/1.4 USM is a better lens for my purposes and was worth the additional money to me, offers some subtle image quality improvements too. But, if you only need occasionally and are just experimenting with a prime (not a zoom) lens for the first time, the "plastic fantastic" 50/1.8 can be a cheap entry lens that might be all you need. Only you can say.
There are lens sample archives attached to this forum... where you can look up image examples and chat about any particular lens. You might want to search that.
So, I'd keep the 70-300mm and supplement it with a walk-around lens. The 28-135 IS would be my personal choice and is likely to be the best overall value (normally sells for upwards of $450 separately). But the newer EF-S 18-135 would be worth consideration. Personally I probably wouldn't consider 18-55 because I use my gear and probably would break such a lightly built lens in relatively short time. I also want faster, more accurate USM auto focus on most of my lenses.
Wouldn't need one right away, but might want to complement it with an wide angle zoom eventually. I would.
It's a toss up about the 50/1.8.... if you plan to shoot candid portraits, it can be a great budget lens for that purpose..... If you plan to shoot a lot of portraits, a better 50mm might be a better choice. If you are thinking this as a "standard" lens (instead of a short tele/portrait lens), then the Canon 24/2.8, 28/1.8, 28/2.8, 35/2 or Sigma 30/1.4 would be a better choice. I use a 50mm a lot on my crop cameras, but opted for the better 50/1.4 instead. The Sigma 50/1.4 is another option, but is a little more expensive and considerably bigger/heavier than the Canon.
Just a thought, I might go for a flash, instead of the wide lens or 50mm, at least initially. Built-in flashes are handy, but kind of wimpy, are in the worst possible place for redeye and ugly shadows, and draing the camera's batteries pretty quickly.
Am trying to keep the difference between the lenses I sell and the ones I buy to a minimum as the wife will kill me if I spend too much more on an already expensive "impulse buy." 
