What size tyres you need will depend on the width of your rims, most rims accommodate a range, but wide rims (which you'll likely have) won't wworkk with anything too slim (and vice versa).
Have a look on the side of your tyre. Somewhere it should have a number moulded into it (you'll probably have to look carefully, it'll be in black on black!), it'll say something like 26 x 1.3 (for example, that's the size of the tyres you linked too) The 26 (inches) is the diameter of your wheel, the 1.3 (inches I think, sometimes it's a number like 28, which is mm, sorry, it's confusing!) is the width. If you get tyres with the same number combo, they'll fit, whether they are knobbly or slick. You might be able to get away with something narrower - if you post what you have on here, someone will know, or a good shop can advise.
As to standing the weight, they'll be fine. It might be an issue if you tried to run on very thin racing tyres, but they probably wouldn't work on your wheel anyway. The main thing is to keep them pumped up well - it's the air that takes the weight really, not the tyre - it's surprising how strong air is! And your existing tubes should be fine - tubes work for a range of tyre widths and if you get something pretty similar, the tubes will cope.
Somewhere else on your tyre, it should have a maximum pressure - probably in PSI (pounds per square inch), maybe a figure around 50-60 or higher and Bar (somewhere about 3-4). This is the most pressure they advise you to put in your tyres, and it's good to get as close to it as you can - it makes riding easier, and it helps prevent punctures. If you have a pump with a gauge, that's good. If not, while you save up for one, simply try and keep them pumped up good and hard, so that when you squeeze them with thunb and finger, they don't really give. You're unlikely to exceed the max, well, I would be anyway.
(If you do need to look for a new pump, a 'trackpump', the sort you hold down with your feet and pump up and down with both hands, is a good thing to have at home, you can get more pressure in. But carrying a little handpump is fine to get you home if you need to reinflate after a puncture.)
Oh, and you know you've got this cycling business licked when you remember, when you put the tyre on, to make sure that the bit that tells you the max pressure is next to the valve, so you don't have to look for it each time. I sometimes remember!
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