Best software for converting to HD MP4 H264?

Could someone please point me toward the best software—ideally something available with a free trial—for converting (or perhaps exporting?) an AVI file into an HD MP4 H264 format? My main concern is ensuring the video doesn't lose any noticeable quality or resolution during the process. I'm on a bit of a tight deadline here, as I really need to have this wrapped up by tomorrow evening.
PS: I did download VLC Media Player, and while I can successfully open the file, I've hit a wall when trying to export it; it seems that specific feature is locked behind a paywall.
PS2: I actually have Adobe Premiere Pro installed, but I'm honestly a little lost as to whether I can handle this within the program itself. Should I select H.264 or MPEG4 under the "Format" settings and then pick NTSC DV widescreen under the "Preset," or should I leave the "Format" alone since my source is currently a Windows Media Video? Please forgive my lack of technical expertise; this is my very first motion graphics project, and I only started teaching myself Adobe Premiere Pro about a month ago.
The project requirements specifically state that the final film needs to be delivered in HD MP4 H264 format.

Thank you so much!

One more thing I was wondering about: what’s actually the deal with the difference between NTSC DV and NTSC DV widescreen? If I’ve already set my project to a 16:9 aspect ratio instead of 4:3, does choosing one over the other actually change the video itself, or is it more about defining the intended screen format—basically telling the software which type of display the film is meant for?

So, I just finished an export in Adobe Premiere using the MPEG-4 AVC format with the NTSC DV widescreen high quality Preset, and now the file extension has switched from .avi to .mp4. However, I noticed something strange: the file size is sitting at a mere 62.2 MB, whereas my other videos are all hovering around 622 MB. Why on earth would that happen? I would have assumed the resolution would stay the same, if not somehow increase, right?

DV (whether you're looking at NTSC or PAL) was basically designed for watching stuff on a TV set, not for playback on a computer—and since we're dealing with an H.264 codec here, there's no way this was intended for a standard TV broadcast. You should just leave the resolution and aspect ratio exactly how they were in the original Windows Media Video file; all you really need to do is convert that AVI into an MPEG-4 AVC. If I’m following your logic correctly, that is. Just grab Format Factory and run the conversion through that—it’s much less of a headache.

Benjamin King2 said:DV (whether you're looking at PAL or NTSC) was originally designed for television broadcasting rather than computer playback. Since we're dealing with an H.264 codec here, it’s clearly not intended for traditional TV viewing. My advice would be to keep the resolution and aspect ratio exactly the same as they were in the original Windows Media Video file; all you really need to do is convert that AVI into an MPEG-4 AVC file. If I'm following you correctly, that is. You might want to just grab Format Factory—it makes the whole conversion process a lot more straightforward.

I tried running it through Format Factory, making sure not to touch a single setting—dimensions, resolution, aspect ratio, everything stayed exactly as it was—but I ended up with a file that's only 110 MB, down from the original 622 MB. It seems like I've somehow lost resolution again, and to make matters worse, the image looks even narrower than the original 4:3 ratio. I'm honestly a bit stumped by it. 😕

Just follow the steps I laid out in that screenshot and you won't run into any issues at all...

I finally did it! I decided to step away from using FF this time around and tried my hand directly with Adobe Premiere Pro instead. This time, rather than hitting "Export" (which just gives me that little preview thumbnail), I opted to select "Queue." It triggered one of those Adobe programs—I can't quite recall if it was Adobe Bridge, After Effects, or maybe Adobe Prelude, but I made sure to grab a screenshot of the saving settings just to be safe. Anyway, it worked perfectly and gave me exactly what I was looking for. Thanks so much for all the helpful advice, 🙂

If you're looking for a solid way to handle conversions without spending a dime, I've found that Any Video Converter is actually one of the better free options out there.

Handbrake isn't even close to being one of the top tier options, especially when you're stuck dealing with the free version...