Since you state that you have very limited knowledge and no practical experience, I would say that you need to start out with sealed (lead acid) batteries. They emit no dangerous gases what so ever.
Once you get more experience with your battery solution you might be able to switch to non-sealed batteries (to some they are know as "float"-style batteries). Non-sealed batteries need a little more knowledge and TLC, but can last longer if they get the right kind of TLC.
But non-sealed batteries will emit some level of (unhealthy) gases while they are being charged. Experienced users will tell you that they have no problems dealing with this gas issue. But I would leave non-sealed batteries to the daring, adventures or experienced crowd.
Sealed (lead acid) batteries are typically labeled AGM or GEL, but ask at the store where you choose to buy.
For any kind of lead acid battery (a sealed AGM or GEL or simply "sealed"), it is also highly recommended that you as a minimum get a voltage indicator, to make sure that you do not over-discharge (over-use) the battery. Over-discharging a 12V lead acid battery (sealed and non-sealed alike) will shorten the lifespan. Likewise not getting it fully charged weekly will also shorten the lifespan of 12V lead acid battery.
When I say shorten the lifespan, it would typically mean that it might last for 2-3 years under "normal" use, but could be shortened to 1 year (or in bad situations to less than 1 year) if it gets over-discharge too often.
So for a beginner lead acid battery, I would say that anything over 1 year of life span is good. And during that year, you will get much more practical experience about the "personality" quirks of the 12V lead acid type of batteries.
Measuring the voltage of a lead acid battery is not the optimal way to monitor it, but it will give you a very good (and cheap) beginners method of how to monitor your treatment and use of your battery.
And it can help you to avoid the biggest "got'ya" issues that all beginners of small house batteries face.
It might be worth considering other battery options.
What you might want to also consider is those USB power banks. Because they use a different chemistry than the lead acid batteries, they do not suffer from the same "personality" quirks that a lead acid battery suffers from.
So those USB power banks, they can not be over-charged, can not be over-discharged, nor do they mind not getting fully charged weekly.
So like it has already been mentioned, for dealing with keeping your phone charged, a good USB power bank might be just right for you.
Unfortunately I do not know of any USB power bank style solutions for a laptop. Otherwise this would be ideal for your purposes.
The closest I can think of, is the Jackery Battery packs. But they have an entirely different price tag.
I would however still highly recommend this type of battery pack to anyone with the limited power needs that you mention, and with the limited experience of how to keep a 12V lead acid battery alive as long as possible, that you also mention.
With all of that said - back to your original questions:
What type of lead acid battery is the best kind for use as a "house" battery (or as some also call it, an auxiliary battery) ?
Golf cart batteries (are typically 6V batteries with a lot of amp hours (Ah) ), batteries for mobility scooters (are typically 12V batteries, but typically with less amp hours than golf cart batteries), deep cycle batteries, and marine-style batteries are all supposed to be better at being used as a house battery, than a regular starter battery. And (in general) I would also say that the order (Golf-cart, mobility-scooter, true deep cycle, marine) can be used as a very general order for longest lasting house batteries.
That is, when we are talking about lead acid type of batteries.
Likewise, the amount of lead (meaning weight) can usually also be used as an indicator to how truthful the labels on the outside of the battery are.
So if one battery claims the same number of Amp hours (Ah) as another, but there is a big difference in weight, then I would be very skeptic how they managed to get to get those numbers, and I would be quite surprised if your every-day use, would give you similar optimistic results.
So, in general, the heavier (lead acid) battery will simply also be able to store more energy.
Golf cart batteries can be sealed, but many are non-sealed types. So make sure you get the sealed, and 12V, type of golf cart batteries. Okay?
Mobility scooter batteries are virtually always sealed batteries.
AGM and GEL batteries are always sealed batteries.
Sorry about the rather long story, but I hope it still makes sense to a beginner 'house battery user'.
