Hi,
DIY Audio is something cool but tubes projects are not for starters. They involve high voltages, expensive components and good wiring/soldering skills.
Opamp, transitor based circuits can be battery supplied, breadboarded and components are super cheap.
Here my part list suggestion
1/4Watt axial resistor
https://www.sparkfun.com/products/10969
or equivalent no constrain here
capacitor
Unless you do filters electrolytics one will be the most used
10µF / 100µF will be OK, go for 16V or 25V ones
100nF ceramic , general pupose
1nF ---> 470nF to do filtering
Pots
10K / 47K will be OK look if knobs are easy to get
Transistors
BC547/BC557 --> general purpose / any brand
MJE253/MJE243--> to build discrete audio power amplifiers /any brand
IC
LM386 super standard audio power amp / JRC have a bit more powerful version, ensure they are DIL package !!!
TL072 popular FET dual op amp /any brand, ensure they are DIL package !!!
NE5532 popular Bipolar dual op amp /ant brand, ensure they are DIL package !!!
few IC socket too, 1N4148 diode are also useful
You can have an idea of what order by looking at various schematics from this manufacturer:
http://www.rolls.com/index.php
like this exemple
http://www.rolls.com/pdf/M_MB15b.pdf
For Arduino hardware side just buy a board, connectors a proto shield maybe, few leds and it will be OK to start.
Olivier