My advice, given you are experiencing problems with your broadband coverage, and I suspect that you want to keep things as simple as possible, would be to get an external hard drive. A terabyte is probably more than enough, but there is no price advantage to going smaller. Expect to pay something in the £55 range. That way you have control over the process and your pictures.
Other Thoughts:
Photobucket is awful. Do not use it. You will go mad(der) trying to fight through all the bandwidth sucking adverts. I used to use it, but no more. The problem with many of the on-line photo hosting sites is that (at least the free ones) do not store all pictures at full size, so you may find that something that was big is now shrunk. The online storage sites don't have that problem.
Cloud services can offer some free storage, but if you start talking about a lot of room for pictures, they may not have enough in your allocation. For pictures it is probably the most secure since their backups have backups, but unless you have a fast, reliable, connection I think it could cause frustration. I have good fibre broad band and found that uploading large blocks of pictures to Photobucket and Flikr could get stalled when one image wouldn't travel. I could come back to my machine after an hour or two, thinking it had been busy uploading, to find it had done just 15 minutes worth and then thrown an error which halted the process.
I assume you have a laptop rather than a desktop computer. I have a medium tower case, so have room for more than one hard drive in there. I have found that having my data (photos etc) on a different HDD from my Windows and programs keeps my data intact better. I have had to do several full format re-installs of Windows, which wiped everything on that drive, but all my data was safe on its own drive. Doesn't protect from theft, electrical surge or my home being struck by an asteroid
but I am mainly interested in protecting against data corruption and hardware failure.
DVDs aren't bad for back up, its what I used before portable hard drives became available at reasonable prices. They work better for storing lots of small files, like photos and documents, than for software. They do degrade over time (there was something on the news about this years ago) so its best to have a rolling schedule of back up if using DVDs, re-back up before the old copy gets too old, not just burning to disk and forgetting about it. There are a lot of sites that discuss how long data will last on a DVD. My own experience has been that I have been fine out to 5 years for photos. They might last longer, but I haven't needed to go back further than that.
Best of luck!