I need to replace my rear cogs. My chain is like a peice of spamghetti right now. I was told it's because my rear cogs are worn. Do I need to replace the front gears as well?

I just upgraded from a 105 5500 series to a tiagra 4600 tripple crank set last year.

Chains need to be replaced periodically because the pins that connect the links wear down with use. As the pins wear, the chain elongates a little bit. That extra length doesn't mesh quite perfectly with the teeth on your cogs and wears them down. So in a nutshell, a worn chain also wears down your cassette or freewheel, not the other way around.

If you replace your chain at regular intervals, your freewheel or cassette will last a lot longer. Your local bike shop likely has chain wear gauges, which are a simple tool. You can also measure wear easily with a ruler since the pins on an unworn bike chain are exactly 1/2 inch apart. Measure a 12-inch section of chain, starting exactly at a pin. If the pin nearest the 12-inch mark is more than 1/16" away from the 12" mark, the chain is "stretched" and should be replaced.

Chainrings tend to wear more slowly because there are more teeth engaged with the chain at any given time to share the load. Do you need to replace yours? We'd need more info to help you determine that. If the teeth on a chainring show an uneven "shark fin" or "saw blade" profile, that's a visual clue that the chainring is worn.

Chain wear should be assessed along with rear cassette replacement. My rule of thumb is the cassette will need to be replaced when two chains have been replaced.
The chainrings on the cranks will generally last at least 3 chains. Although there are bike forum members who are still using chains that are original to their 1968
Super Whammo at 50k miles, most of us find chains last 3 to 6 kmiles, depending on cleaning, lube and environment. Get a chain checker to assess chain wear
although a 12" ruler will suffice, with 1/16" elongation in 12" being the lower limit of acceptable chain wear and 3/32 to 1/8 being way too much.

On a triple chain ring, you generally spend 50-70% or your time on one ring, 15-30% on another ring and 5-10% on the other. The most used ring will most likely
be the one needing replacement depending on how you ride. Both cassettes and chainrings wear the same way: the teeth get ground away, on the front side on the
chain ring and back side on the cassette. When the wear reaches a certain point the difference in spacing of the teeth and the chain causes the chain to ride up on
the teeth and "jump" causing a noticeable clunk in the drive train under torque.

There is no doubt that the chain is worn. The guy at the shop used that tool and it's supposed to stop in the middle, the tool didn't stop at all. The chain moves side to side like a piece of rope. I get a clunk at the lowest front gear. And one of the teeth on that gear broke. I had to file it down beacuse it was catching my chain.

The front is a new tiagra 4600 triple. The rear is a used ultegra 6600 10 speed i got from ebay last year. The chain was a brand new Chain I got last year. The guy at the bike shop told me there is no point in getting a new chain because the back is so worn but if the chain is supposed to be replaced from time to time (something I did not know) maybe I should just pop on a new chain.

I will check for the shark fin and saw blade visual cues.

If you pop on a new chain, and the cassette cogs are worn out, you will get "skipping" when you pedal in the most worn gears. My guess, based on your info, is that you need a new chain and cassette. The front chainrings are probably ok.
Chains wear faster than cassettes, so if you replace the chain before it's completely shot, you can keep the same cassette at least until you replace the chain a 2nd time. Depends on how well lubed and clean your drivetrain is.

1st two pics are of worn chainrings, last is of one not worn.

Finally got some time to take the back gears apart. They look like they still have life in them to me. I think i am just going to pop a new chain in. Then the next time around I will replace the cassette. Actually I will probably move on to a new bike by that time.

That 12T looks kinda wasted, kinda hard to tell, though. modern cogs have pretty funky shapes even when new.

I've worn out plenty of chains, cassettes, and a few chainrings. I have *never* been able to visually detect wear on a cassette, even when it was worn out to the point of causing the chain to skip.

That makes me feel better because I can't tell for the life of me if they are worn or not

here is 12t up close and personal although it might be the shadow and grease marks that make it look like that.

Yeah they are worn. The new chain is hopping like crazy under tension when I shift to the middle gears.

Now I have a decision to make.

1) replace the cassette, and return the cassette I purchased which doesn't fit my wheels.
2) replace the rear wheel which does have a slight hop that I was told is impossible to fix and use the cassette I already have.

EDIT: decided to just stick with my current wheels. I got a replacement cassette from amazon.