First, find out why running makes your knees sore. Otherwise you're going to just keep injuring yourself and might have to walk that run portion. Unless you can somehow manage a 30 minute bike split and run/walk the 10k in less than an hour then you don't have to worry about run training.

I found out I had knee problems once I started running more than a few miles once I got into Tri's (never really did much distance running before that) and finally figured it out when I went to see the doctor along with a sports medicine physical therapist.

What kind of history do you have with running? If you are basically just starting out soreness is to be expected. It will take months of consistent training to become accustomed to running Do you have any prior knee injuries from the past?
I would start by running/walking every other day for a goal of 30:00. Alternate running and walking and over time shorten the walk segments until you are running 30:00.

I used to suffer a lot with knee pain. Long cycles would do it, but running was hugely problematic. It appears to all have come down to knee stability. Some of the postural knee muscles just weren't sufficiently engaged. I saw a physio who diagnosed the issue and gave me some strengthening work to do. It gradually resolved the issue. It occasionally returns if I've had a layoff from running (cycling doesn't seem to provide much useful stimulus to these muscles) but gradually fades as I get strong again, especially if I do some more stability work.
Your best first step in my opinion is to get an appointment with a good physio. It may well be something that can be addressed relatively easily with a little advice and some work at home by you.

I don't know how you run but If you're heel striking your knees are going to suffer no matter the volume.

Thanks for the feedback! It sounds like I need to get to a physio/doc to check out my knees first, then slowly ramp up the running.
As for getting to the doctor, this probably seems like a goofy question but should I go straight to a knee specialist, or setup appointments with physical therapists?

As for my running background, I have very little. I am active, but not much running at all. I cycle several times/week and walk my dogs 40 minutes every evening. Otherwise I sit or stand at a desk all day. I believe the longest run I've ever completed was only 10k!

I think the issue is just mainly you have no real running to speak of so Im guessing its just soreness from not doing it on a regular basis rather than a knee problem.
Ive been running for 34 years and I had to take off 6 weeks recently from running because of an ankle injury. After resuming again just after 6 weeks off I was sore everywhere for a good 2 to 3 weeks and it took me a couple of months to regain what Id lost.

Just go straight to a physio. A doctor will add no value. It's probably nothing of consequence. If the physio has concerns, they can refer you.

Might be muscle imbalance from being a cyclist.

I've been doing strictly Du's for several years. If I spend a winter with a heavy focus on cycling I get knee pain in early spring when I start up running again. Same thing vice versa when I've parked the bike for a 4 month marathon build, except it plays havoc on my Achilles when jump on the bike. Cycling and running engage the quads and hammys differently and when they're out of wack for the activity you're working on pain will happen. (How's that for a technical breakdown.)

Worth seeing a specialist to be sure. Hopefully they just need to point you towards some strength building exercises to kick start the re-balancing.

Shgock absorbing insoles might help to stop shock from attacking your knee.

Maybe but I would advise the exact opposite approach!

If you're just starting running I think it's a perfect opportunity to use very little cushioning so that you avoid developing bad habits that will cause problems later with or without shock absorbing insoles or highly cushioned soles. It may not be worthwhile for established runners to try and retrain themselves if they've always used heavily cushioned shoes but it's nearly as easy to start running in more minimal shoes as heavily cushioned ones and they will almost certainly lead to better technique and less injury in the long term.

I transitioned to minimal shoes when I started back to running after various problems. I realised my technique was just rubbish (adult onset runner) and this seemed like a sensible way to address that. I've never regretted it for a minute. My technique has changed completely and I enjoy running much more now.
If you run right you don't need shock absorbers, and they really don't add any comfort either, just weight and reduced feel. I've done an Ironman in Merrell Bare Access with zero problems to feet, or knees. On the other hand I had lots of problems with aches and pains, sore knees, sore back and achy hips when I ran in heavily cushioned shoes that encouraged me to heel strike all day long. A small bit of cushion isn't a bad idea for long distances just to protect your feet a little, but I'm talking 6-12mm not 25mm.

Most running injuries (some studies put the number as high as 80%) relate to weakness in the adductor muscles, or abductor tightness... Most of the time you get nondescript pain in the kneecap area it's some degree of PFPS (Patello-femoral pain syndrome). This usually relates to muscle imbalance. Cyclists tend to be prone to adductor weakness, since you're often not recruiting a lot of those muscles in your pedal stroke... Spending some time in the gym (either home gym or proper gym) to do some basic strengthening will go a long ways towards reducing this. Also, proper stretching, foam rolling and massage to help address any abductor tightness (ITB, piriformis, gluteus medius especially) will also go a long ways...
Supplements are a waste of money for dealing with this, it's just good old school hard work...
In terms of the running itself, start slow and build gradually... It's worth seeing a physio to ensure that there's nothing else going on. If the impact is bothering you, in the initial phases of strengthening things up, you can certainly pool run...
In terms of shoes, it's a complete fallacy that anyone with a history of knee problems needs to go more cushioned... You need to look at your stride, and make sure you find a shoe that is suited towards your mechanics. The tradeoff of additional cushioning is lack of proprioceptive feel, in many cases (especially when pain is linked to muscle imbalances) extra cushioning can do more harm than good. Getting something that provides any needed stability, but not extra stability, and that has adequate "ground feel" will give you the best ride in terms of being able to detect deficiencies in your stride and correct them. It's also a good idea, to incorporate regular form work into your training, to help prevent bad habits forming...