Here's my two cents on the topic after messing heavily with pedals for more than a decade.
1. Tuner. That's easy. Two choices here.
Either go standard with one of the classics (TC poly, Korg Pitchblack, Boss TU-3) or go fancy with the Peterson HD or the Sonic Research.
A tuner is essential - it allows you to tune your instrument and it doubles as your mute switch. Very handy.
2. Compression: My only tip for a beginner would be to leave compression alone for the time being.
This is a vast topic, but a comp is a very subtle effect and its usefulness is not immediately evident. If you use it to even out your playing dynamics it can become a set of training wheels that will actually make sure you never learn to balance on your own.
3. Dirt. There's dirty boosters, overdrives, distortions and fuzz pedals. Each is a rabbit hole. Each is highly subjective. You can loose yourself into finding the perfect pedal for the job - and then you discover that you can stack dirt pedals: Welcome to RabbitHole². However, these are simple. Buy a bunch on the second hand market. Plug in, twist knobs and either find what you like or don't. Keep it in the former, sell it in the latter case. Keep in mind that there is a huge difference between your bedroom tone and your tone in a band mix, though.
4. Modulation effects: Chorus, vibrato, flanger, phaser etc.
These have a tremendous effect on guitars and are less pronounced on bass, especially in a mix. Yet, they all have their uses if you can bring the use case. The best way to get into those is to start off with a cheap multi to see if something really works for you.
5. Time based effects: Mainly Reverb and Delay.
I really dig Reverb. I bought one when I thought I wanted to play guitar more, but it went on my bass board. I love just a hint of reverb as an always on effect. I also like to have big ambient reverbs. Delay certainly has its uses - but IMHO, there's a learning curve to delays, too.
6. Preamps:
This is a bit of a complex thing, because the borders between categories are a bit fuzzy here.
A preamp can be as simple as a box with a baked in sound and minimal controls (Chase Tone Secret Preamp for example). It can also be a box with no baked in sound whatsoever and a simple or complex EQ and DI out (like the Tech21 Q\Strip). Or something like the Sushi Box Underground Accelerator.
Preamps can (sometimes) shape your sound like an EQ. Preamps can (sometimes) add dirt. They often have a DI out. You can do multiple things with them in multiple parts of your signal chain.
7. EQ/Filter:
In my personal book, the difference between EQ and preamp is that the EQ tries to alter certain frequencies after the fashion you set with their knobs while most preamps will color the sound.
Filters would be divided into static ones, mostly low pass and high pass - and dynamic ones, like envelope or LFO controlled filters. The static ones are super simple. They cut off what's below their threshold.
In recent years, high pass filters have become very popular - and for a reason. It is counter intuitive, but setting a high pass filter a bit above the root frequency of your lowest note can actually help your lows sound better.
A flexible EQ can be a great help on a complex board.
8. Power Supply and cables:
What I learned here is that you should buy a bigger psu than you think you need, and that you should buy a good one, with isolated outputs. The same goes for cables: Don't cheap out on them. Buy ones with small plugs that won't protrude too far out of the socket to save on real estate.