In respect of the Tesco box and 'no signal' many people don't understand the way Freeview boxes work, and this may be your actual problem. If you have used it at one location and then move somewhere else you will have to make it do a retune. However for many STBs that will just be a scan to see if it can find any additional channels or stations; if it already thinks it has a BBC1 and finds another BBC1 it may put it in the 800's or it may ignore it. You need to do a full (a.k.a. factory) reset (there will be such an option somewhere in the menu structure) to wipe its memory, then get it to do a new scan. It will then find the correct channels for wherever you are at that time.
Also remember that only main transmitter sites and some of the important relays will broadcast all channels; smaller relays will only transmit the 'usual' 1-5 channels plus a few others, but on such sites you will be missing things like ITV3, ITV4, PickTV, Yesterday etc. Relays will transmit PSB1-3 also known as BBCA, D3&4, and BBCB (which is HD only.) Have a look at the following link to see what stations you can expect from a relay.
http://www.dtg.org.uk/industry/dtt_channels.html
To confuse you more..... there are channels and stations. Channels are like we have now on analogue where each station (BBC1, ITV, etc) occupies one frequency. However in the digital world each channel carries a multiplex (or mux for short) where each mux contains up to eight TV stations and maybe ten radio stations, hence much more efficient use of spectrum and why DSO is happening.
After Digital Switchover (DSO) is complete next year your tv/video will not work. Most such sets are only able to record from an off-air signal and as it is analogue - and there will be no analogue - it is defunkt. If the set has a SCART socket on the back you may be able to use the TV to watch programmes sourced by a STB but is suspect you will not be able to record. Now is the time to get a new TV - there are some superb bargains to be had - but buy a good brand (such as Samsung, LG, Panasonic) as the cheaper ones just don't last. If you have a John Lewis' within range they usually offer a free five-year guarantee which is well worth having: if you don't look at Richer Sounds as not only are they price competitive but they only charge 10% to extend the guarantee to five years. Avoid the likes of Curry's or Comet as they will charge you a fortune for an extended guarantee.
There are plenty of digital recorders but they come in two forms.
The Personal Video Recorder or PVR is the most common around and the cheapest. Most have twin tuners so not only can you record two programs on different stations at the same time but (if you have a digital TV) can watch a third, or you can watch a recorded programme whilst the other two are still being recorded. You can also time shift: say a one hour prog starts at 20:00 but you won't get home until 20:15, you can start watching the recorded programme at 20:15 (say) whilst the PVR is still in the process of recording the original. A PVR records on a computer-type hard disc so there is no simple way of retaining the original, and you have to delete progs once you have watched them or the hard disc will fill up.
The second option is a sort of PVR with DVD recorder, usually known simply as a DVD-HDD recorder. On these you can record directly onto a DVD giving you a permanent copy, or you can record on the hard disc (like the PVR) and either watch and erase, or transfer it to a DVD if you want to keep it. These type of machines usually only permit one recording at once even though some have twin tuners so that the second tuner can be used like a STB to feed the TV.
For PVRs the cheaper end is such as Goodmans, TVonics, Sharp and supermarket brands; the more expensive end include such as Humax, Topfield, and Sagem. You will see Philips around but they are winding down their TV operation - not to say that they don't make good products so there may be some bargains to be had. If you want decent quality look at the Manager's Specials at
www.humax-direct.co.uk
Don't worry about complexity - almost all of them are easy to set up, in fact most will do it for you. If you run into problems there are plenty of places on the InterWeb where you can get help.