Lead acid batteries should always be recharged after use. Alkaline batteries can be left flat. So first you have to consider will you take the battery back to house to charge after each use. This determines which type of battery to use. Lead acid are easy to charge but often alkaline need special chargers. It does not matter is NiCad, NiIron, or NiMh all do not need keeping fully charged but the Lead Acid does again does not matter if open cells or valve regulated.
As I am sure you realise lead is heavy so the largest battery you can consider carrying is around 110 amp hour these are made as leisure batteries and leisure batteries are designed to be cycled i.e. 50% discharged the recharged again on regular basis. The car battery would have a very short live before lead falls off the plates. Because of carrying the valve regulated battery is likely better they come in two types gel and absorbed glass mat but for you it does not really matter which. Even a 7 Ah will cost £19 so these are not cheap.
Using LED lights is not a problem even a small inverter I used a 150W inverter to run the lap top but as the inverter size increases so does the battery size. You should not really aim at less than an hours use so 7 Ah will give around 5 amp for an hour yes I know 7 Ah but that's over 20 hours. So 5 x 12 = 60 so 60W is your limit. You may be able to run 150W with odd burst but not for long and inverter would soon trip out with under voltage.
7 Ah is a common size so cheap at £19 but jump to 12 Ah and more expensive mobility scooter batteries are about the best bet
here are some examples with prices unlike the cheaper caravan battery they don't leak. But first you need to consider size of power tool. A 750W drill would likely work with a 26 Ah battery at £47 for the odd hole but would not really last that long. You will also require a charger and inverter some inverters also charge.
Inverters come in basic two flavours. Simulated sine wave and true sine wave. My laptop works great with simulated sine wave which are cheaper, a standard electric drill will likely also work OK but with a speed control in trigger likely it will need true sine wave. True sine wave tend to be better made small inverters 150W or so with simulated sine wave there is no problem but larger ones I found from bitter experience tend to have a short life the FET's which work them need to power share and that seems to be the problem. The true sine wave are better made and seem to work far better.
However as the cost goes up one quickly reaches a stage where a generator will do a better job and is cheaper. These also come in two flavours those with inverters and those without. Without an inverter the motor runs at a set speed all the time and noise is a problem. With inverters they can tick over at light load and only rev up when being used. Some inverter models can also be paralleled up but this is near impossible with non inverter models.
Using a small lead acid for lights great but really power tools it's generator which is required. With small solar panel you can leave a battery at garage which will re-charge for lighting only. Size of solar panel required to re-charge with power tools becomes silly.
I would look at caravans and boats to see what can be done with battery power. Visit a caravan or boat place and see what there is.