Batteries seem to be an area where there is conflicting and often misinterpreted information.

To put it simply, the Volts is the power of the battery (how fast it will turn a motor, how brightly it will light a lamp etc).
Amps is how long it will do it for. Unless you're using a lot of amps a big battery (in Mah)is no faster.
It generally corresponds to how long the device will run.
From my limited experience a 2Ah battery will give my about 30-40 minutes of normal "playtime" on my tanks.
A normal "stick pack" that comes with your tank is about 1800-3000 Mah and is usually an arrangement of 6 x 1.2v per cell NiCd or NiMh. Nominally 7.2V for the pack.
Whilst they work fine a LiPo Lithium Polymer pack is much more efficient and compact as well as offering slightly more voltage (3.7v per cell) 2 x 3.7=7.4v.
Using batteries any of higher voltage in a 1/16 tank is a recipe for disaster. The circuits and drivetrain are designed around the 7-8v paradigm and so all of the electronics, and mechanical components are optimised in these ranges. I've done a 12v system for giggles and used a tank mixer with car ESC's. Stripped gear boxes and a thrown track was my reward. It was fast for a while though.
I'd imagine 12v and 30-40A esc's for a 1/8 or 1/6th scale.
Our basic 1/16 tank in standard form draws about 1.2amps per side flat-out along the ground with 500gms of ballast. Up hill and in stall conditions this would go up considerably but they really don't use much, my simple run time observation from above seems to concur.
So, our actual POWER is... 7.5v x 1.2A = 9 watts per track x2 =18 watts total. The ancillaries use so little as to be inconsequential.
There is no definitive as everyone drives differently and batteries are not all equal.

Do not bother even keeping NiCads. The technology is so old and their care is so fussy it's just not worth it.
Remember the "memory effect" this is exclusive to NiCads and their energy density is pitiful. I began RC racing in 1983 and that was all we had so I know all of the pitfalls of a a NiCad and there are many.
NiMh cells have had their day in the sun, I still use them and they are a good idea for the standard tank config.
Slow charge rates and bulk are the biggest disadvantage but for all intents and purposes they work well. (remember they are still 1.2v per cell x 6 = 7.2v) They usually only come in the standard "SubC" rendered a standard by the RC car racing community back in the '80's
I'll drop in a truism here that my father imparted many years ago...

"A new item does not render the old one incapable of still doing its job"
So there is nothing wrong with using those old batteries lying around.

Lipo's are here to stay so this next segment is probably of the most interest.
Personally I see the biggest advantage is the ability to use different configuration and sizes to allow for chassis layout optimisation.
A 2.5Ah LiPo with a hard case is smaller than a packet of cigarettes, (not a politically correct simile but I'm sticking with it).
Personally I have been using 18650 cells, they look like a AA on steroids and are the most common LiPo cells in the world.
Most laptop packs are 18650 cells in a series/parallel arrangement, likewise most power tool packs. This is why we see common voltage ranges in most tool marketing (a lot of manufacturers round up to 4.0v per cell as a fully charged LiPo is about 4.15-4.2v).
Elon musk uses these by the 1000's in modular arrangement to drive his legendary TESLA's.
So there we have a cheap source for LiPos without breaking the bank. Most laptop and tool packs fail because 1 cell dies, some manufacturers put a cell count and the battery dies whilst still being perfectly good.
We are not going for hyper performance here, our use is mild and if you can obtain the cells from recycling then you have an ample supply of cheap cells.

Lipos also feature another specification that seems to confuse people
The infamous "C" rating.
Simply put a batteries capacity means the it will deliver its rated voltage at that current for an hour.
The "C" rating is a multiplier of its current that it can deliver continually.
So...
A 7.4v 2500mAh 10c battery will deliver 10 times the rated current in our example 25amps continuous for 6 minutes.
Using the same formula 7.4v 2500mAh 40c battery will give us 100amps for 90 seconds. This is important for racing cars and planes as often a large current surge is needed on acceleration of in the case of planes stunt manoeuvres.
Now you might understand why we often see 80-100 amp ESC in cars and for a 1.2m metre scale warbird a 60-80A ESC is common. They need to deliver very strong bursts of power but again how you drive/fly can drastically effect the runtime of any of these devices.
The effect of not having sufficient capacity or "C rating" feels like the battery is flat. It cannot deliver the required power.
Our tanks have relatively no dynamics in the drivetrain and so a 2000mAh battery of about 8C is all that is really needed.
Having a better rated battery will not hurt but any hi performance LiPo is completely redundant in this application, a battery of 5C is sufficient run a standard chassis.
I'll follow up with some example photos soon (Later tonight if I can).
I just bought some new Battery cradles and will do a tutorial on fitting a 2 cell 18650 holder into a standard chassis, a cheap and effective very simple and cheap upgrade for a standard unit.