The university publications will include watering instruction.  Those drippers are more than enough, you'll adjust the soil moisture by adjusting the timer.  Look at university publications for advice on putting the soil together.  I would have used a mixture of top soil, manure, sand.  Maybe a little perlite.  Test the soil to determine whether it needs to be amended to adjust pH, you can get a kit at any store that sells gardening supplies.

Lay the tubing out, it will relax in the hot Sun down there.

OP your plants need more nitrogen. (based on the pics before the carrot plant in your hand)

You have wood chips on that garden, and although I have not read closely, I'm guessing you are watering.

As the wood chips rot, they tie up the nitrogen in the top layers of soil, and therefore it is not available to your plants.

So...you need to fertilize with a VERY MILD solution of something like Miracle Grow, or the equivalent.

NOTICE...I said VERY MILD.

That's important.

If you are new to gardening, don't make the mistake of thinking "if a little fertilizer is good, more must be better."

it's not better.  You can kill plants with too much.

Mix it half of what the box says.

Three days later, look again.   Go easy with it.

ETA:  Most carrots are about 80 days to maturity from planting.  So at 57 days, you have a while to go.

Lettuce:  You want young, tender leaves.  Start pinching off the mature leaves now and see how they are. they should be tender and sweet, not tough or bitter.  You don't pull the whole lettuce plant (if you planted leaf lettuce.  Head lettuce is different.)

Peas...what has your weather been like?  What temperatures?  Have you seen blooms?  Are there insects to pollinate the blooms?  (don't know where you are in AZ, and don't know your climate for gardening, but peas like cool "spring-like" temps.

1-Peas are a little paler(or sometimes just a "different" ) green than some other crops.  Your peas look good to me.  Yes, the pods follow the blooms.

2-Those onions do look a little spindly.  See #4, below.

3-That lettuce color looks pretty good overall.  A tiny hair darker green wouldn't hurt it, but spring lettuce is usually a lighter green, depending on the variety.  I would keep doing what you're doing and take that lettuce as a good job.

4-If I'm seeing the photo right, you may be overwatering just a tiny bit. (It may not be as wet as it looks to me.  I want so badly to reach in the screen and feel that growing media. ) You don't want your soil to dry out to the point that plants wilt, but the plants don't need to be in wet soil all the time.  That may be part of the trouble you're having.  Maybe skip a day watering (or wait a day longer between waterings) and watch it closely and see what happens.   Especially with root crops, you don't want the roots too wet.

You have a conundrum.  Lettuce loves more water.  But your onions and carrots cannot have "wet feet" and do well.  Drainage is key for them.  So don't back off too much.  But maybe just a tad bit.  Watch carefully and take everything I'm saying as from a gardener who has never grown anything in your climate.  These are general thoughts, so keep that foremost in mind.

If you stick your finger down in the middle to bottom of that soil, is it wet?

That "cool and wet" ....may be a key.  I'm not saying you should get your soil bone dry. Certainly not.  But properly drained soil should not feel wet to the touch all the time.  The best of the watering world from the plant's point of view is to drink nice and deep, dry out a little, drink nice and deep, dry out a little.  When the soil is wet all the time, the roots cannot get oxygen (roots do in fact breathe, after a fashion).

Now then....that's in a greenhouse, or here in Kentucky, and works in all regions where I have ever functioned as a gardener.  But I have NOT functioned as a gardener in the desert southwest.  Just sayin.  If letting the soil get a little dryer before you water again would cause your plants to get too hot or stressed, don't do it.  But my guess is that they won't, and may well benefit.

I think you will be okay.  If it's JUST mold on the growing media, it won't hurt anything.  If you hate the looks of it, rake it away and dispose of it.

Glad you cut back on the watering some

Very cool!  I think gardening of any fashion-putting your hands in dirt and planting seeds and watching something happen that we, as humans, cannot make happen, is good for people on all levels.  Body, soul and spirit.  Especially veggie gardening, because then you eat what you grow, and the cycle continues.

I will now shut up about my philosophy of life, but I think it's cool that you're bringing the kids into it.  They will never forget, I guarantee it.

Those are aphids to my eye, and they do come in black as well as several other colors.

Are they ONLY on the lettuce?

You actually don't need to do anything to the soil proper.  You have mulch, if I remember right, so I would lightly rake the top layer of mulch to disrupt anything hiding in there (insect larvae I mean) but I doubt you have much.  You will fertilize same as usual when you start to grow again.

One question...

Have you been eating regularly out of the garden, or had it been sitting there for a while before you went out to get this last bit of lettuce?

Aphids are not particularly hard to control but they can be a pita.

You have a lot of approaches.  Everything from introduction of beneficial insects to the environment, to a hard spray of water (this is temporary but does discourage them) to insecticidal soap (probably your least chemically dangerous choice) to the bigger guns.  My advice would be to start fresh and have some soap ready for next year.  Beware...soap is not "if a little is good more is better."  You can burn your plants with soap.

I think, also, that thinning the lettuce and not letting it stand there too long will be of benefit to you.  This is a "don't attract them in the first place" method that may or may not work.  I don't know your growing climate as I've said before, but if they arrived ONLY after the lettuce was up and on its last legs, it may be that you can do a fair bit of insect control by just staying on top of the garden cleaning, etc.  This is beneficial in a lot of ways, mainly by interrupting the pest's reproductive cycle by removing them and/or depriving them of what they need when they need it.  This style of gardening is OVERWHELMINGLY SPECIFIC to the climate and microclimate where you garden.  What will work for one will likely not work for the neighbor across town with slightly more shade or a windier spot, and vice-versa.

I'm suggesting it because it's good practice anyway, and because your pests seem to have come right at the end of the viable garden life of the lettuce.

If you search the web, you will find gazillions of articles and posts on aphid control.  Hard to know what's decent and what's not.  I used to suggest Gardenweb without  fail, but Houzz bought them and they've messed up the site.  I would suggest at least looking at the advice of your state's extension service, and finding out what's working best in your climate. They will default to chemical control, probably, but there will be good information about time of year, damage to crops, etc.  It will be good learning.

Kitties

OP, if you haven't considered it already, in the next growing season, you might consider starting a small compost operation.  You can do this with something as simple as a smallish metal trash can set over to the side, out of sight.  It doesn't have to be a big expensive deal.

It might be a really nice adjunct to the garden for your kids, and you could compost not only the garden trimmings/trash at the end of each growing cycle, but veggie scraps from the kitchen as well.  Then your little ones get to see the food they grew (and even what you bought at the grocery and did not eat) return to the earth as the best fertilizer available on the planet.  Black gold.

Back Yard Science.