Multiple things:
1. Pictures of the leaking caps in question, please. Best with ratings visible.
2. Try something easier for starters, if you never soldered before. I used some LED blinking solder kit to get started, then worked my way up.
3. You can probably order correct replacements from digikey, mouser etc.

Also, are those caps through hole or surface mount? For through hole I can only recommend using a desoldering gun if you want to replace more than a few caps, because otherwise it is very time consuming and dangerous (lifted pads, damaging components etc.)

This guy here has some of the best soldering YouTube videos I've watched. Check out his various videos, because his techniques/methods are proper 95% of the time. https://www.youtube.com/watch?v=fYz5nIHH0iY

For 100% proper soldering, follow IPC 7711B/7721B. I got IPC rework certified through work and took the training -- I highly recommend it if you can get your work to pay for it, which they should since it would be a professional development course. (if you're working in a engineering/tech related field) Otherwise, you might be able to find a book or materials online. If not, I can try to upload the scan of the IPC 7711B/7721B content somewhere.

EEVBlog is a great source. For pure solder technique, without comment and chill music, check out Androkavo:

https://www.youtube.com/channel/UCKDfmGeSKmwP6SdGDHhu6hg

Here are the caps. They are through hole.

I have replaced caps on a motherboard before, so I have a very basic idea of how to do it. (Not saying I am good at it).

My main question is whether I can replace cylindrical polymer caps with radial aluminum electrolytic caps, in general.

Thanks.

Failed polymer caps are relatively rare. Visibly leaking ones are even rarer I'd say. Those look like polymer caps, but they may not actually be. I'm not that experienced in this area but generally if they stepped up to polymer caps, there was a reason, so I'd definitely replace them with similar caps. Still... Seems very odd to have them fail like that if they truly are polymers.

A picture of the whole card, and the brand name may help too.

Edit:
Aha... I think I was right to be suspicious of those.
http://www.capacitorlab.com/capacitor-types-polymer/
They are standard wet electrolytic caps, they just don't have a sleeve. Polymers should have completely smooth tops with no vents at all (except for certain yellow ones I guess?). I'd never really thought about this before, so I'll have to make sure to distinguish these from polymers in the future. Normally polymers have odd capacitance and voltage ratings too.

Ozzuneoj wrote:

Failed polymer caps are relatively rare. Visibly leaking ones are even rarer I'd say. Those look like polymer caps, but they may […]
Show full quote

Failed polymer caps are relatively rare. Visibly leaking ones are even rarer I'd say. Those look like polymer caps, but they may not actually be. I'm not that experienced in this area but generally if they stepped up to polymer caps, there was a reason, so I'd definitely replace them with similar caps. Still... Seems very odd to have them fail like that if they truly are polymers.

A picture of the whole card, and the brand name may help too.

Edit:
Aha... I think I was right to be suspicious of those.
http://www.capacitorlab.com/capacitor-types-polymer/
They are standard wet electrolytic caps, they just don't have a sleeve. Polymers should have completely smooth tops with no vents at all. I'd never really thought about this before, so I'll have to make sure to distinguish these from polymers in the future. Normally polymers have odd capacitance ratings too.

Thanks!

tegrady wrote:

Here are the caps. They are through hole. […]
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Here are the caps. They are through hole.

I have replaced caps on a motherboard before, so I have a very basic idea of how to do it. (Not saying I am good at it).

My main question is whether I can replace cylindrical polymer caps with radial aluminum electrolytic caps, in general.

Thanks.

What's the card? Can't find who manufactures that series, but they might not be polymer. Some are wet electrolytic in a polymer-like casing.

They look suspiciously like Sacon FZ caps, but I can't be sure atm.

gdjacobs wrote:

tegrady wrote:

Here are the caps. They are through hole. […]
Show full quote

Here are the caps. They are through hole.

I have replaced caps on a motherboard before, so I have a very basic idea of how to do it. (Not saying I am good at it).

My main question is whether I can replace cylindrical polymer caps with radial aluminum electrolytic caps, in general.

Thanks.

What's the card? Can't find who manufactures that series, but they might not be polymer. Some are wet electrolytic in a polymer-like casing.

They look suspiciously like Sacon FZ caps, but I can't be sure atm.

Yes, I no longer believe they are polymer caps based on the previous post.

Also, it is an apparently generic Geforce2 MX400 128mb card. I cannot find a manufacturer's name anywhere on it.

gdjacobs wrote:

[...]

What's the card? Can't find who manufactures that series, but they might not be polymer. Some are wet electrolytic in a polymer-like casing.

They look suspiciously like Sacon FZ caps, but I can't be sure atm.

Second that. That casing may be polymer-like but these are clearly failed electrolytic caps.

It's almost the new version of fake cache: fake solid/polymer caps. Fake and clearly unreliable too.

The caps are most likely for noise suppression or buck converter use. Poly caps, 6.3v and 1000uf with similar case sizes will probably work fine (not guaranteed, though). Still, a generic GF2MX might not be worth it. I'd give you one of mine if you were close -- I've got a few extra.