If you're a teenager, get a PSP slim - if you're an adult, get the PSPgo - but really, any PSP is good for emulation.
I have 2 PSPgo's, a couple of 1000's, a 2000 and a 3000. People often forget about the PSPgo but personally, the PSPgo's are my absolute favourite for several reasons - I'll buy them whenever I see one. They have 333MHz clock-speed, 64MB RAM, 16GB internal storage, and can fit external SD storage memory. The built-in flash ROM is super fast for emulation and with high-speed USB, it takes no time to transfer ROMs to it. They have a better screen and better battery life, lovely form-factor and design and the case can be modded with "armour" 3D print to give it a huge battery capacity (2xbatteries) and a Playstation controller grip. It has 2 memory slots (built-in flash-Rom and external SD card) - although you'll have to buy a ribbon adapter and tape it inside the casing. Wifi, Bluetooth, Ad-hoc (supports 16 consoles for multiplayer), built-in microphone. The whole thing slides open beautifully and slides closed to fit in your shirt pocket. It's by the most expensive PSP, but after my fist one, I've had around 4 of them in my lifetime - gifted 2 and kept 2. Absolute worst is PSP Street for many reasons (screen, "touch" controls break, less RAM) - but hey, even though it's technically the worst choice, it's still a good system and looks/plays fairly decent.
I sold the 1000's and never touch the PSP 3000 now, it seems bulky - I'm thinking of taking it apart and putting it inside an iCade casing to make a small MAME cabinet. PSPgo will take only 2 minutes to install "infinity" CFW. The only complaint you'll ever see about a go is that they are "uncomfy" to hold - I found that not to be true - it's different, but not uncomfy - but when you get the armour casing with the PS3-style grip, it's actually better than any other PSP for comfort. It has a video out socket to link to a TV (PSPslim has a TV out too). If you get an upscaler, you can get HD PSP games on your TV with it. Both of mine have the battery mod (original battery+LiPo high-capacity battery), 16GB internal flash memory, 128GB external, the ribbon mod, screen protectors - one is black, the other is white. Regardless of which PSP you go for, use the custom firmware to overclock the MIPS CPU to 333Hz for emulation, set the Card speed to "normal" for games and "Fastest" for menu. This will speed up the SD card when browsing your games but not cause issues when playing games.
Use .cso format (level 9 compression) for your PSP games and PSP mini games, use eboot (pbp) format for your PSX/PS-1 games. You can also get most PS-1 games down to just a couple of Megabytes by ripping the video/music/trailing-zeroes and packing files from them to save loads of space. Use PSP content manager to sort your lists and manage your games. I have 1000 PS1 games that take up less than 60GB on my PSPgo's SD card. Don't bother with N64 emulation at all - but you can run MAME4All, FinalBurn, NeooGeo, TurboGraphX/PCengine, GBA/GBC/GBX, SNES/NES, MegaDrive/MasterSystem, C64/ZX Spectrum, Amiga, ScummVM, Atari flawlessly - and you can even run a few DOS games using DOSbox (you need to trial-and-error them all to manually see which ones work properly). There are also a few decent PC ports (Homebrew games) which you can grab from the internet archive.
If you don't want to dump your own UMD games, EmuParadise is still the best place to get ROMs (use tapermonkey and a workaround script to enable downloads). If you want, you can also edit the games to add your own music/art/icon/screenshots for browsing the PSP menu. A final tip is to change the PSP theme - most of them are rubbish but some of them are lovely. Basically, it's personal preference. The "best" PSP is the one you love to play. Go to a CEX store (cash converter/pawnshop) and have a look and feel of them before you buy one. Whichever one leaps out at you, get that and dedicate your time into modding it - they're all decent machines. You can sometimes grab them really cheap at garage sales/craigslist/gumtree etc. In a few months, you'll probably expand your collection with a different model anyway. Be aware, you can also use a mid-range android phone/tablet with a bluetooth joystick to emulate PSP and PSX games perfectly - another option is a PS-vita with adrenaline installed.