One consisted of two convex lenses of different focal lengths and the
other one consisted of a concave lens as the objective lens and a
convex lens as the eye-piece(that's called a Galilean one I guess).
That's the ABC of refractor design - the most basic types of refractor. They are only interesting from a historic perspective.
You can certainly purchase a large convergent lens (bi-convex, or plan-convex) and use that as an objective. Then, for the eyepiece (ocular), you can use either a small convergent lens (bi-convex, or plan-convex), or small divergent (bi-concave, or plan-concave). Probably only the first one will work well: the image will be inverted, but the field of view will be nice and wide. The second design (divergent lens for eyepiece, Galilean refractor) provides a non-inverted image, but the field of view is extremely narrow and hard to use; avoid it, it's frustrating.
However, this sort of bare-bones design is only interesting if you want to reconstruct the history of astronomy. At least for the eyepiece you could just "steal" any cheap 1-1/4" eyepiece from a telescope or microscope. Even the cheapest one will work much better than your single-lens "eyepiece".
For the objective, again, I suggest you don't use a single convergent lens. It will work, but it will be full of aberrations (image distortions, color fringes, etc). Instead, go on Surplus Shed, click on the Lens Finder. For lens type, choose Achromat. For diameter, anything over 45 mm. For focal length, anything over 190 mm.
An achromat is a combination of two lenses, glued together, that are designed to reduce aberrations. The image will be much better.
Their stock varies greatly. Right now, I've found these two:
The 50 mm diameter, 300 mm focal length achromat, combined with a 15 mm focal length eyepiece, will provide 300 / 15 = 20x magnification. With a 5 mm eyepiece instead, it will give 300 / 5 = 60x magnification. Don't go much higher than this (more magnification is not always better).
The focal length of the objective dictates the length of the telescope tube. E.g. with the 300 mm achromat, the telescope length will be 300 mm plus some extra at the back for the eyepiece.
EDIT: Be mindful of the f-ratio. Basically, divide the focal length by the diameter, and that gives you the focal ratio.
A lens with a diameter of 50 mm and focal length of 500 mm has an f-ratio of f/10. A lens with a diameter of 50 mm and focal length of 300 mm has an f-ratio of f/6.
It's best to use an achromat with an f-ratio between f/8 and f/20. If you go below f/8 it will be very hard to collimate (fine tune) the scope. If you go above f/20, it will be too long.
Sometimes you find super-cheap eyepieces there, too, like this one:
Also keep digging around that site, maybe you'll find various other parts necessary for building the refractor.
Enormous care must be given to keeping the objective and the eyepiece collimated (aligned and centered). Even a tiny collimation error will greatly affect performance. This is not a trivial engineering problem.
There's also an easier way. You could purchase a complete kit, like the Galileoscope:
It's a 50 mm (2") f/10 refractor that has about the same performance like good binoculars. The price is very affordable (and very cheap for what it does). As a telescope and mirror maker, I was impressed by the quality of the achromat doublet objective that they provide (50 mm diameter, 500 mm focal length, f/10). For the price and the size, it is well made and well corrected, and punches well above its price category.
The kit comes with everything you need: all the lenses, the tube, various small parts, etc., including components for the ocular (eyepiece). Another nice thing is that you can swap its eyepiece for a standard 1-1/4" eyepiece used in a regular telescope (like a cheapie on Surplus Shed). Whoever designed this thing put some good thought in it. There are several different instruction documents on the site, that you can choose from.
My 9 and 12 year old sons put it together in an afternoon. It's that easy.
EDIT: What can you do with the Galileoscope?
Obviously, you can watch some of the planets. You'll see the two big equatorial belts on Jupiter, and the 4 galilean moons of Jupiter. Saturn's rings will be quite obvious. Venus will show a crescent - it will look just like the Moon, only much smaller.
You can watch the Moon, too. The big craters are quite visible in an instrument this size.
You can watch some of the deep space objects too. The Orion Nebula is quite visible actually. The Andromeda galaxy is somewhat visible from a place outside the city.
Double stars such as Albireo, which appear as single star to the naked eye, are easily split by a 50 mm aperture. Albireo is very pretty, and easy to see in a Galileoscope - an orange/blue double, easy and wide, very beautiful.
Mizar, one of the main stars in the Big Dipper, is also a double. This one is a lot more tight. I think you should be able to split it in a G-scope anyway.
For the planets and the Moon, you'll probably want high magnification. 50x or more, up to 100x (the maximum supported by this instrument).
Make sure the instrument is sitting on a stable, firm support. A flimsy, shaky tripod is the enemy of high magnification.
Look on Surplus Shed and other online stores for 1-1/4" eyepieces. A 5 mm eyepiece will give you 100x magnification - the maximum. A 10 mm eyepiece will give 50x. A 20 mm eyepiece will give 25x - nice and wide, good for big objects like the Orion Nebula.
Look for eyepieces with a wide field of view (at least 50 degrees, up to 80 degrees, or more, if you can find them). A wide field makes everything easier.
It is also possible to build everything from scratch, including making the lenses and/or mirrors yourself from a piece of glass. But that is a MUCH bigger commitment, and a much more long-term project (a few months to a year).