Instead of a phono stage you can use a linear amp like a microphone amp and apply the needed RIAA correction using software.
Software like Pure Vinyl allows you to work this way.

Recording

A lot of debates about bit depth and sample rate can be found on many audio forums.
According to Shannon-Nyquist your sample rate should be the double of the highest frequency.

Vinyl can contain signals up to 25 kHz (but higher values are reported) so 44.1 kHz (Nyquist=22) is probably a bit at the low site.

When doing the AD conversion any signal in excess of half the sample rate generates an error.
The input should therefore be band limited.
Low sample rates like 44.1 kHz forces you to use the same steep filtering (brick wall) as in the first generation CD players.
Higher sample rates allow using a smoother filter.

 

Vinyl can have a dynamic range of 70 dB (but often limited to 50 dB) so 70/6=12 bits should do. We can hear certain tones to 30dB below the noise level. So 70 + 30 dB = 90dB.

16 bits should do the job.

But you need some headroom and in case of post-processing it is beneficial to have a 24 bit format.

 

Ripping vinyl is a hell of a lot of work.
Even if 24/88 is overkill as some advocate, I rather would go for the overkill.

 

Some recording software can be found here.

 

RPM

There are many things you can adjust with a turntable.
One of the most obvious things to check is the speed.
Use a strobe platter  to do so.

Obvious I must be old recommending a strobe, because there is an app for it.

 

RIAA

Because of the vinyl revival, youngsters buy a turntable only to discover it sounds horrible.
What they don’t realize is that you can’t connect a turntable straight to a line in.
You need a phone input.
Not only to bring the weak output of a TT to line level but also to apply the RIAA correction.
In a nutshell, because the groove cannot accommodate large amplitudes, the bass has to be lowered.
Because vinyl it self has a rather rough surface, the treble is boosted to improve the SNR (Signal to Noise Ratio).

 

On playback, the opposite is done.

RIAA was implemented as a standard around 1954.
Before the time, each record company applied its own standard.
Lovers of 78 rpm recording know this.

Needle

78 rpm records were made of shellac and had a wide (mono) groove.
This limited the playing time to 5 minutes per side.
Later (Columbia, 1948) they switched to vinyl (PVC) and a much smaller groove increasing playing time to 23 minutes per side. This became a industry wide standard.
Cartridges at that time had 2 needles. You could simply flip it.

 

 

AD converter

If the turntable has analog out ,you need an AD converter to digitize.

This is typically a sector dominated by pro-audio companies like Prismsound, Metric Halo, etc

Older models often have a firewire interface.

Newer models support USB 2.

 

Click and Pops

There is software to remove clicks and pops.

This requires you to make a needle drop.

There are sulutions to do it in real time

 

Basically boxes like these have a AD converter, some program doing the de-clicking and a DA converter.
Indeed, a mini PC with a sound card can do the job as well
 

Real Time Click Repair MiniPC Setup - Steve Hoffman Music Forums

 

Vinyl archiving – Werner Ogiers
An article covering aspects like AD-converters, sample rate, LP production process, recording software.

Guide to Converting Analog Vinyl To Digital Files Using Windows - Mitch Barnett

How vinyl records are made - Do it