I used to fix VCRs as a teenager (when the cost, and my income, made them worth fixing), so I always had enough VCRs to record anything I wanted.
You might need an amplifier for an antenna signal, but a cable signal can often be split 4-5 ways into modern VCRs with little picture degradation. Tuners today are pretty sensitive. Use a single five-way splitter, because every single connection on the signal path causes significant signal loss. Also, use cables with screw-on F-connectors instead of the slide-ons often packaged with VCRs.
You could, in principle, “daisy chain” 2-3 VCRs in series between the source and the TV (i.e. connect the output of one into the input of the next), but sooner or later you’re going to forget to set all the VCRs to “TV” (the “TV/VCR” switch could more accurately be called “Tuner output/direct signal (bypass)” but that would confuse more people than it would help.
I’d recommend a 5-way splitter on the signal, splitting it to the VCRs in parallel, and a manual pushbutton 5-way switchbox between the outputs of the VCRs and the TV. you need a video amplifier, place it before the splitter. For casual viewing you can use one good VCR for playback, and several cheap $30-40 for recording additional channels. The TiVo can be treated as a VCR in this case.
This may sound extravagant, but total cost could be under $30 for cables, splitter and switch plus $30-40 per VCR. If you want an amp, I’d recommend a good distribution amp with a built-in splitter to 4-5 outputs. They can run as little as $30-$50 on eBay. (I’m afraid you’ll have to read up on the brands; neither cost nor brand familiarity are good guides. The familiar consumer brand names generally market terrible and overpriced amps for about $30 while the best and most cost-effective brands will seem like no-names to consumers because they don’t market to, and don’t pretty up their packaging and cases for, the end consumer.)
Personally, I think a $15 cable crimper, a pack of spare F-connectors, and 50’ of quad shielded RG-6 coax are essentials for any household, but I’m weird. Making and repairing your own cables is vastly better than buying/replacing commercial cables. It’s really easy, and you’ll get better cables than most consumer cables that people use. (there are some great prosumer cables out there, but they can be pricey: a single cable can cost more than the crimper, connectors and raw cable needed to make 10 cables at home, and your wiring will be far neater if you make you cables to length, instead of using 6’ cables when you only need 1’
A brief word on dB: this is the unit used to measure signal amplification or attenuation. Though the subject is more complicated than even many electronics technicians realize (because the formal definitions dB in signal voltage, power, etc. are NOT straightforward, and consumer spec sheets rarely tell you which standard is being used), I can crudely say this: the decibel is a logarithmic unit. a 10 dB gain or lost is a tenfold increase or decrease. a 3dB gain or loss is a doubling or halving of the signal. (This is not always strictly true - e.g. with video signal voltage, 10 dB is more like a factor of 40- but it gives you a general idea of how dB works: adding a few dB cuts your signal strength dramatically. When you add or subtract dB, you’re adding/subtracting an exponent!)
But be careful. You might just find yourself with a hobby that’s much more useful and engaging than watching TV.