The previous answers have already pointed out that spears are indeed used in the show. The same is true of the comic books. I won't explain what's going on here, because it would be an enormous spoiler, but the image conveys all the necessary information pretty well.
The spears are manufactured by the Hilltop colony and used at the Hilltop and in the Kingdom.
But They Didn't Use Them At First
However, in the comics, spears as such (meaning a long wooden shaft topped by a large arrowhead-like object, as opposed to any long, pointy thing) only appeared in the series at a relatively late date, when issues related to ammunition supplies became a serious problem. In the show, it only took about 2 seasons for some sort of spear (i.e., the long pointy kind, not the actually-a-spear-by-design kind) to show up.
Why didn't people start using spears earlier? My guess is that they had some trouble acclimating themselves to the new reality in which they found themselves. We're used to the idea of using guns to kill stuff. We haven't used spears as anything other than equipment for an obscure Olympic sport (the javelin throw) for quite some time. We tend to solve problems by using things we are familiar with. We are all familiar with guns, whether or not we've ever used one1. We are less familiar with spears, and we think of them as ancient, outdated weapons used by primitive cultures and Roman centurions and stuff.
On a similar note, we generally solve problems using items that are readily available to us at the time. Most people in 21st century America don't have spears laying around all over the place. If you wanted to have a spear, you would have to make it yourself. This isn't a big problem if you're in a semi-permanent settlement somewhere, surrounded by high, strong walls, and populated by plenty of like-minded people. But if you're out on the road fending off starvation, dehydration, bandits, sickness, zombies, and who knows what else, you don't have time to start manufacturing spears and practicing how to use them.
Finally, humans have an unfortunate tendency to ignore imminent threats until they are right up in our faces (kind of like we ignored climate change until it was too late to avoid or mitigate). The threat posed by running out of ammunition is no different. As long as people had a reasonable amount of ammo, they were happy to keep using it.
The obvious problem with this is that there are no more factories churning out boxes of every kind of ammunition imaginable. The number of bullets in the world is now finite, and rapidly shrinking. On top of this, the first thing people did when zombies began to eat everyone in sight was run to the local gun store and beg, buy, borrow, or steal every last round of ammo in the place. Then, for the most part, they took their booty of ammo, found a place that seemed safe, and promptly died. So all the ammunition that used to be concentrated in a relatively small number of locations (like gun shops) is now scattered in tiny caches all over the country.
So people ran out of ammunition pretty quickly - within a year or two of the world falling apart - and only then did they stop to think about what they would do without it. And that is when everyone realized that guns were great, but should be used sparingly, and more primitive weapons were the new rule of the day.
Assuming that the show continues to follow the general course of the comic books, you can expect to see a lot more of this sort of thing - improvised weapons, old fashioned weapons like bows and arrows, spears, swords, axes, maces, etc. And you'll probably see some other examples of people falling back on tried and true earlier technologies - for the first time in centuries, castles are starting to look pretty awesome and useful to the few remaining survivors. Even an old fashioned frontier fort/stockade, with walls made from upright logs, is a very attractive option.
Why didn't they start using spears sooner - the TL:DR version:
- It wouldn't have occurred to them very readily, because no one uses spears nowadays.
- Conversely, they very quickly focused on getting ammunition and guns, because those are the most familiar weapons to modern people.
- They didn't have readymade spears handy.
- They didn't have the time, know-how, resources, or inclination to make them, or an environment safe enough for them to focus on making them, until they found semi-permanent settlements.
- They didn't think about what would happen when they ran out of ammunition until they had almost run out of ammunition.
1 Keep in mind who the characters are: Americans from the southeastern state of Georgia. When the series begins, the group is led by two cops. All cops in the US carry guns, and in Georgia, somewhere in the range of 40% to 51% of all households own at least one firearm (as per the map below; other sources say the figure might be as low as 32%). Just to the west, in Alabama, gun ownership is even higher - from 51% to 63% of all households. These are people who are likely to have grown up around firearms, and statistically speaking, about half of them probably owned one or more guns before the inevitable zombie apocalypse began.
Gun ownership in Georgia is broadly reflective of gun ownership in the country as a whole: About 1/3 of Americans own guns, and most gun owners own more than one firearm. Thus, we find the oft-cited statistic that there are more civilian-owned firearms in the US than there are people:
In short, Americans are, on average, more familiar with guns than is the norm in much of the Western World, and southerners are more familiar with guns than most other Americans. As such, it is entirely to be expected that, when the inevitable zombie apocalypse comes, most Americans (and most southerners) will think of guns first, and other weapons later - probably only after ammo supplies are running critically low.