What kind of bike?

What do you mean by 'train rides'?  Are you referring to group rides or some other type of riding (for example off-road)?  What type of triathlons are you planning on doing - off-road, on-road?  What is your budget? That will help people know where to start in offering suggestions.  Also, do you currently have a bike now?  Is so what kind; and what do you like/not like about it?

Train rides?  "Training" rides or rides on a commuter train?  Hybrid as in a commuter bike or a cyclo-cross?

Personally (and this is just my opinion), if you're looking into competing either in a road bike race or a triathlon, and since you're already in the market, I would look harder at the road bikes.  Eventually, you'll want to step up to a road bike anyway, so if you've got the money and inclination to buy now, buy into a decent first road bike. 

I think the OP probably meant "trail" - as in a hybrid is more versatile to go off-road. But Cami bee is right - we need some more info. What are you riding? How much trail vs road/paved trail riding do you plan on doing? Do you think you want to stick with triathlons, etc? I know we usually tell folks here "whatever bike you have is good enough for your first race" - which is true. But if you are buying a NEW bike, meaning you already have one, I would say to get a cheap road bike if you plan to race. You don't need to drop a grand or whatever - even look for used bikes. Just make sure that it FITS you. Depending on how much you are looking to spend, you can probably get a used road bike for not a whole lot more than a hybrid. And if you already have a mountain bike, etc, a hybrid, imho, would be a waste of money. If you are going to do a lot of road riding and triathlons that are not off-road, a hybrid is going to be considerably slower than a road bike. But again, we need some more info.  

The Walmart bike should be fine for cruising around with your kids. I would invest in a better bike than a hybrid, to be honest. If you search hard enough you can probably find a used road bike for a few hundred bucks. It may not be all fancy and tricked out but it'll be a lot faster than the hybrid in race situations. Again, just make sure it FITS.

My advice then is to keep at this until you decide if you like it or not.  If you buy now, two things will happen.

1. You will like doing this and you will want to buy a better bike.  You are then out the 3-400 bucks..
2. You don't like like this sport and you will not want to ride the bike anymore.  You are then out the 3-400 bucks.

Good point. I guess I should expand and say, wait and buy a better bike when you know you want to keep doing this. Don't waste your money on a hybrid if you end up wanting a decent bike.

How is the Wal-Mart bike uncomfortable? Is it the seat, the reach, etc? It may be fixable at least for the time being. You don't want to train on a bike that hurts to ride.

I think the cyclocross bike is what happened when Lance Armstrong designed a go-between bike.

 If I could only own 1 bike (and keep in mind I own 4) it would be a cyclocross bike

X2

A hybrid is fine if you never intend to do any serious biking, and it that's the case then you would be just as well off with your Walmart bike. Everyone I know who first got a hybrid ending up getting a more appropriate bike within a year thus wasting their money. Decide what type of biking you are going to primarily do and get the proper bike. Many hybrids fall into the category of "comfort bikes", meaning they're heavy, slow and put the rider in a position where going fast is almost impossible - but they will be comfortable!

You can also run in Walmart running shoes or the slightly better shoes from places like Kohls. Just don't expect the shoes to provide the same type of support or performance as proper running shoes.

The point is that if you're going to do tri, get what is necessary for a successful experience. Getting inappropriate equipment is a recipe for struggling and never realizing your potential.

In addition to the highlighted quote: If you are going to use a running program regardless if it is for tri, marathon, 5ks, etc., a properly fitted running shoe may prevent injury(ies) and discomfort. There are folks who give up on running because everything hurts all the time. Proper shoes may have prevented this from happening.

A local running store should take their time to watch you walk and run and in each shoe they recommend based on your answers and their observation. Note: "Anytime during your first few weeks of training you may return the shoes and we will replace them with a more appropriate pair". That is what you want to hear.

Edit: As far as the bike. Any bike can ride on a trail and the road. However, it is easier to do it on the appropriate type. If you truly want a new bike for trail and road, test ride a cyclocross and mountain bike. I had two wheel setups for my Mountain bike. Trail tire and road tire setups. Otherwise, save the cash until you are sure what you want to do and ride what you have.

Best of Luck

I agree.  Most hybrids can't handle off road use, they are meant more for light bike riding, in an upright position.