>>>If it use it at home, will I be able to wifi connect to my unlimited Sky broadband?
All tablets can connect to any convenient wifi signal. So you could use a 'wifi only' tablet at home or anywhere that there's free wifi access. (e.g. all large Tesco and Asda stores, all J D Wetherspoon pubs, Caffé Nero, many railway stations, most B&Bs, Wagamama and Giraffe restaurants, and many other places). You could also use paid services (as long as you were happy to fork out for them), such as BT Hotspots, on-train wifi, etc.
The download speeds you get with the wifi services from Tesco, Asda, Wetherspoon's etc are generally quite good. (Typically between 2Mbps and 12Mbps whenever I've tested them). However 3G and 4G services use signals from mobile phone masts. 3G has limited data capacity, which is probably why you find the internet access from your phone is quite slow. Since a 3G tablet uses the same system, you might not find it much quicker.
Any device using mobile phone signals requires you to pay for the service, which is why you'd need a data plan. It might technically be possible to 'tether' a tablet to your mobile phone, so that you could use your existing data plan, but that might be outside the terms of your contract'. You really need a separate data plan.
Many people with 3G tablets find that they never actually use the 3G facility, finding that free wifi access is good enough for their needs. (Some people might also pay £6 per month to get access to BT's Hotspots all over the country, so that they can get much wider wifi access). If you think that you'll really need lots of fast internet access (and you don't mind paying for it) you should consider using 4G [which is much, much faster] from EE.
http://shop.ee.co.uk/ipad-tablets-group/