how much time each week varies greatly by number of critters and the efficiency of your setup. If you have auto waters and feeders with good fences and pasture, you might get by with very low time inputs during a routine week. If you have the pigs in a small pen that needs to be cleaned, fresh water trucked in daily, cleaning waterer and feeder daily, or bad fences, you could spend an hour or more a day on just a couple of pigs or goats.
I have chickens that take two visits daily. The first visit is to let them out to free range, check water, and collect eggs and takes all of 5 minutes. Then there is another visit after sundown to lock the coop and takes about one minute. Two pigs in my set up take not much more than 5 minutes a day unless I have to run to the feed store. The two horses take longer because their water troughs take 15 minutes to fill.
I raise both hogs and goats, and like the previous poster said, time depends on your setup. It also depends on what your end goal is.
Are you looking to buy weaned animals and feed out to butcher? Or raise from conception to consumption? Time involved varies greatly for the two.
Hogs are nice in that you generally only feed one feed, are relatively low maintenance, and have a quick turnaround time to feed out, maturity to breed and gestation. Sows can also produce 16-20 pigs a year (if breeding 2x per year). They usually require assistance with birthing (even if it's just someone there to make sure piglets don't get laid on). They are also usually bred using AI, so you don't need a boar. You can order the semen. However, they are hard on fences and facilities and can be challenging to handle (loading and such).
Goats are much easier on fences and structures, and much easier to handle normally. However, they usually require more space (if on pasture) and access to forage and grain. They grow slower, have a later maturity for breeding and a longer gestation. They also usually only produce 1-3 kids a year, but don't usually need assistance birthing. They are more challenging to AI, so many are bred using natural service, which means access to a buck is needed.
In terms of other products, hogs are pretty much just for meat. Goats can provide meat or milk, depending on what breed you get.
I love both, and enjoy the challenges each species provides. I have around 40 goats and spend probably on average 30-45 minutes a day with them. I have 6 sows and spend probably 10 minutes a day with them.
Before you decide which species to raise, determine what facilities you have or need, access to feed resources and cost, access to vet care and what your purpose is with them. Then you can see what fits best.