Just to give you a tip-of-the-iceberg oversimplified answer, a lot depends on what the formats are, and also why you're converting them....
a) are the native camera "formats" digital or analogue, and are they on tape, or are they in files?
b) are they hi-def or standard-def?
c) are you converting them for playback only, or for editing and/or eventual conversion to another format (eg, editing in a NLE (non-linear editor) and perhaps for eventual conversion to a playable
d) are you planning to archive this material? For possible later re-editing, or in finished form? How do you plan on storing it?
e) professional or personal use?
The answers to (c) and (d) can get particularly complex, as when you ask any editor what "format" he/she would like, they will always tell you that they'd like to have it in the native format or "codec" of the NLE they're using. This turns out to be an ever changing target, and no one I know has come up with a universal answer. For more on "codecs", see links at the bottom of this post.
There are lots of issues in going "cross platform", some simple once you're aware of what they are, some not. Some Apple formats, eg, are proprietary, and many professionals are not confident that Apple will maintain support for them. There are also Apple-PC issues, and there can be aspect-ratio issues.
I will take the liberty of mentioning what's generally considered to be at the top of the line encoding/transcoding software for most professional uses, and that's Adobe Media Encoder, which comes bundled with Adobe Premiere Pro, Adobe After Effects, and Adobe Prelude software. I don't think it's available standalone. It's not cheap.
Recommended hardware for Adobe Premiere Pro is:
Intel Core2 Duo, 64-bit, 8GB of RAM, preferably RAID drives.
Adobe Premiere Elements is scaled down from the Pro version, and is roughly $100. System requirements are less. The Elements Encoder also less versatile with some file formats. I imagine it's adequate for most consumer cameras, but I don't know. I'd guess from your questions that the Elements Encoder might be what you want, if it supports the formats you need. There are also several utilities that I'll discuss with the mod about whether I can post.
Regarding hardware... for many personal uses, some performance glitches might not matter. A friend who outputs his Nikon DSLR video to FCPX routinely shows "dropped-frame" messages, but the video looks generally OK and the dropped frames don't seem to matter, though they might affect sound sync.
Analogue tape will require a hardware I/O device as well as software to encode.
Also, there are lots of other "encoding" and "transcoding" video conversion utilities, some free, that posters are liable to drop names to, and for the time being I think it's probably best not to discuss those... until we clear with the moderator here. Cheap codecs can be a problem, and some utilities include codecs that can conflict with other codecs needed for editing. Playback only and editing codecs may not play well together, but I'm not precisely sure of that. Generally, the free utilities convert between playback formats, not between intra-program editing formats.
Archiving digital files on hard drives is considered problematic. As I posted on this forum several years ago, it was a problem that hadn't been solved... and it still hasn't been solved. More on that later.
Some professionals are continuing to maintain digital clones of the "last stable digital tape format", as well as to hold onto original film, as a way of future proofing valuable material, but machines and people to run them become harder to come by, and climate-controlled storage is expensive.
I've been in the process for several years of coming up with archiving procedures. Am currently in crunch mode on this as my media vault recently shut down. I've gotten my most important digital tapes and backups separated, in relatively climate controlled conditions (ie, inside, not in a humid warehouse, and in separated buildings). Frequent cloning is recommended, and that can get onerous and expensive.
Scoping out a computer that will handle uncompressed standard def and moderately compressed hi-def has been an ongoing project for me... perhaps fortunately interrupted as my requirements have changed.
For more on codecs, here's the standard introductory article... FAQ: CODEC's, a Primer by Bill Hunt
Jul 31, 2012
The article refers also to these two Wikipedia articles... Codec
...] Video Codecs
More later, but this should get you thinking about requirements, whether you want to pass videos down to great grand kids, etc.