Need Lens Purchasing Help

Question

Hi folks,
So first ever post and relative newbie to the world of photography. My issue is this. Currently using a 6D and own the following glass: 70-300 f4-5.6, 24-105 L Kit lens, and 85 1.8. So i feel like I'm covered for most of my type of shooting (portraits, street, daughters outdoor sports, etc). BUT...when my daughters are playing indoor hoops and competing in indoor gymnastics, I feel hamstrung. The 70-200 2.8 would be perfect, but at $2500 is just too expensive. I thought about replacing the 70-300 with the 70-200 2.8, but I feel like i might miss the former too much when they are playing outdoor sports or if I'm looking to shoot wilderness shots and not use a multiplier. 
Sooo, my question to you guys is a) would you make the swap i outlined above, b) do you have any suggestions as to other lenses to consider, c) is the non IS 2.8 worth considering as it is much cheaper?
It should be noted that I currently handhold most of the time so the weight of the 2.8 may be an issue. But let's assume I pony up for mono pod in the near future. 
Anyway, sorry for the lenghty first post, but I've struggled with this for a month and need help. Thanks for any thoughts you guys (or girls) have. 

Answer 1

Someone already mentioned most of the suggestions I'd make...
70-200/2.8L IS Mark II would be tops. Sharp enough to make your eyes bleed, even wide open. But you're right, it's rather pricey. Some alternatives...
70-200/2.8L IS "Mark I", still a great lens if you can find one used.
135/2L IS is a super lens, a full stop faster than f2.8 and very fast focusing. It would complement your 85/1.8 very well, too. (I consider the 135L a "must have" portrait lens with the full frame cameras, but it's also an excellent sports lens.)
Yes, the 200/2.8L II is another super lens, if you need the additional reach on your 6D. An alternative is to get a quality 1.4X teleconverter to use with the 135L, to make for a 189mm f2.8 lens. I use the Canon 1.4X Mark II and it works very well with the 135L.
6D is able to shoot at really high ISOs with little loss... if the lighting in your indoor venues allows, the 70-200/4L IS might be worth a look.... it's also about 1/3 smaller and lighter than the f2.8 versions if weight is a concern. Doesn't come with a tripod ring, tho, so if you want one it'll be at some extra cost.  
If it were me, I'd probably go with any of the above before the 70-200/2.8 non-IS... that's the oldest of the Canon 70-200 lineup and probably has the weakest image quality of the bunch. Not that it's bad... It's just not quite as good as the others, IMO. Wide open I'd rank the 70-200/2.8L IS Mark II and 70-200/4 IS the best in terms of IQ, as well as the latest and greatest IS, the rest slightly behind... emphasis on "slightly" tho. I'm also a big believer in IS, after using lenses with it for close to 15 years now.
I don't see any of the above replacing the reach of your 70-300mm. However you might consider eventually replacing that lens with something else, possibly even longer for field sports and wildlife/birds if those are what you mean by wilderness shots. I use the EF 300/4L IS a lot, it's very good both with and without 1.4X teleconverter, especially on a full frame camera, and not much different in size and weight from a 70-200/2.8. Most other long tele lenses will be a little or a lot larger and heavier. There are the Canon 100-400L (similar size), Canon 400/5.6L (a little bigger,very sharp, but I wish it had IS), Sigma 120-400 OS, Sigma 150-500 OS, Sigma 50-500 OS (nicknamed the "Bigma", a clue to it's size and weight), and the recently introduced Tamron SP 150-600mm. There are others, but the above are the more affordable choices and prices for the others tend to be significantly higher.  
Some sample images might help with your decisions:
300/4L alone on 5D Mark II (ISO 800, 1/8000, f5.6, handheld.... Top image is cropped from the original for composition, center AF point was on the bird's eye. Below is a more than 100% detail from same image) ...
Normally I wouldn't use such a high shutter speed even for a fast moving bird in flight, but I was shooting in the shadows and didn't have time to adjust my settings for the above shot when that guy suddenly flew close by... I spent 45 minutes trying to get some more good shots of that bird, since it continued to hunt close by, with better settings and other cameras and lenses, but never got another that I felt was as good as the first one! 
Now here's a shot with the 300/4L IS + 1.4X II teleconverter, also on 5D Mark II (ISO 1600, 1/250, f8, monopod used partially for self-defense, but thankfully not needed for that purpose.... Top is very slight crop of full image, below is a detail from same image tho I'm unsure of the magnification)...
Next, below are a couple shots made with the 135L mentioned above...
This one with the 135L on an APS-C crop camera (7D, ISO 1600, 1/125, f7.1, handheld)...
And this one on full frame like your 6D (5DII, ISO 6400, 1/200, f2.0, handheld, very low available light)...
I also have tons of shots with 70-200/2.8L IS "Mark I" (below image made with it on 7D, ISO 800, 1/800, f5.6 at 140mm, handheld)...
And with the 70-200/4L IS (also on 7D, ISO 800, 1/640, wide open at f4.0, handheld, avail. light)...
As to the weight of the 70-200/2.8... well I'm just shy of 60 and most of my exercise involves the TV remote... yet I often manage to shoot all day with a 70-200 on one camera and similarly sized 300/4 on another (as well as a bag and vest with a couple smaller lenses, flash, water bottle, extra batteries, lots of memory cards, etc.)... Sure, there are a few ibuprofin tabs involved at times, but it's very "doable".
But if you can, go check out the lens in a store to see if you think you can live with it. If so, great! If not, the 70-200/4L IS is a bit smaller and lighter alternative, as is the 135L (though both are still very solid feeling and well made, like most L-series).
Have fun shopping!
Alan Myers
San Jose, Calif., USA
"Walk softly and carry a big lens."
GEAR: 5DII, 7D(x2), 50D(x3), some other cameras, various lenses & accessories
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