Ammonia Reading

Hi,
I've got a fairly new tank that I had to transfer my fish to due to an aggression problem with another fish, so I had to do some moving around. I had wanted to take my time getting this tank cycled and ready for fish, but unfortunately I've had to hurry along in order to protect another fish.
I've taken one of my sponges from a mature tank, along with several gallons of mature water and put into the new tank. I'm also using Tetra SafeStart.
Last night, before I put the mature sponge into my filter, I did a nitrite and ammonia test and the readings were both at 0.25. So then today I did a water change and added the mature filter sponge, as well as some of the water from the mature tank.
I've just taken another reading and this time my nitrite is reading 0, which is good, but my ammonia is still reading 0.25 -- is there a reason for this? And should I be worried? I know basically that ammonia and nitrite should both be at "0", but with this being a new tank, I know I'm going to have spikes. I just want to know if there's anything more I can do, or need to do?
I've been keeping fish off and on for the past 30+ years, but I will admit that my knowledge of water problems is very limited. I used to just ignore the problems, hence the reason my tanks never lasted very long. I only just got back into fish-keeping, on a more serious level, last year, and now I keep mainly puffer fish. I must be doing something right, I've got 53 puffers (9 different species) and haven't lost any yet, other than one getting sucked into the venturi on the filter, and another 3 dying from whitespot (Ich).
Anyway, any help or suggestions appreciated.
Thank you!

Thirty years ago we didn't think much about the nitrogen cycle and beneficial bacteria. In a new aquarium, it is not unusual to have ammonia and nitrite. That's because the nitrogen cycle is not complete. Since the cycle is powered by beneficial bacteria, you want to stimulate the growth of a healthy colony. The filter media from an established aquarium will help. If you have some gravel from the other tank, that will help, as will live plants. You can also add a little bit of fish food every day. If you already have a fish in the tank, you will need to do frequent, partial water changes until the ammonia and nitrite get down to zero. The bacteria colonise the gravel, filter and plants but not the water.

A few months back, one of the fish magazines said that floating plants like hornwort and water sprite will directly metabolise ammonia. I tried it (lots of hornwort) in two new (used, old, free) aquariums, one a 46 gallon tank and the other a 30 gallon. From day one, ammonia and nitrite were at zero. After a month or more of getting the same results with different test kits, I added fish They're still healthy.

That's very interesting about the Hornwort, I will look into that. My nitrite is at zero, but the ammonia was reading 0.25 last night. I will check it again in a little bit to see if it's gone down. My husband cleaned our other external filter from an established tank and I took some of the water and poured it into this new tank, so I'm hoping that has helped the situation some.
Thanks John for the suggestion, I appreciate it.
And just to say, 30 years ago I must have been an idiot because I didn't do any of the things I do now with regularity and I wonder how I kept any fish alive at all. I remember taking out all the ornaments and rinsing them in tap water, and I can't imagine that there must have been much beneficial bacteria in those tanks! Funny how one day it just snapped in my mind that I needed to do more and needed to learn from other people.
I keep puffers, which is a little more involved than most tropical fish, but I still tend to be a little stupid when it comes to other areas of fish-keeping, but at least I have some place to come to when I have questions and concerns.
Thanks again to everyone.

Hornwort is easy to grow, but you need a large bunch of it to get it started.

PS: My first aquarium was in the early 1950s, and I had a subscription to one of the popular aquarium magazines. So far as I can remember, the nitrogen cycle was never mentioned. In fact, they didn't talk about it at aquarium clubs either. Experienced aquarists would advise letting a new aquarium sit for a week or two before adding fish. Most likely, nobody was aware of the nitrogen cycle or its importance. Or maybe it was just the naive kid in me.

Until coming here to England, I can't remember anyone talking to me about "cycling" or "water changes", and I didn't have any idea about nitrite and nitrate, although I had basically heard of ammonia, but didn't know it's importance relating to aquariums. I think I just stuck the tank in the corner, fed the fish, cleaned it when things went wrong and that's it. I feel so foolish when I think back how I used to keep a tank. I don't think any of the pet shops ever asked me about my tank, or if the fish I was buying were compatible with the fish I already had.
I feel like I've been reborn now that I know the real deal to keeping fish!