You will need a processor board, motor driver, sensors, and audio amplifier.
Since you are asking on the Arduino forum, it would be natural to consider one of the Arduino boards or derivates - but beware they tend to be rather antiquated compared to alternatives now available. The plus side of the classic ATmega boards is the wide variety of read-to-go example code available right within the IDE, and the wide supply voltage range (of the bare chips if not the boards) which is more compatible with AA-cell battery packs than competing device's narrower voltage range tends to be. The one part of the functionality you describe that is a bit of an issue is the audio output - you can get it from an ATmega but it will be poor in quality and you will need to add storage for the "bark" data. You might almost consider a separate module with an ATtiny and a small SD card for that (ironically the ATtiny while generally less capable has fast PWM better suited to audio).
For motor drivers, especially when using batteries steer clear of the older bipolar L293/L298 type of solutions, as these have very high internal loss which basically means that one cell in your battery pack exists just to generate heat in the driver IC. A modern FET based driver (the TB6612FNG is quite popular) will be better.
For sensors you probably want one or more of microswitches extended with "whiskers", ultrasonic rangefinders, or short range IR reflectance.
Some dual wheel drive (or "tank steering") RC toys can be worth considering as platforms, but their gearing may be too fast for your purposes, particularly those designed to spin and tumble. Continuous rotation servos are sometimes mentioned, but it can be hard to get them tuned to not rotate when commanded to stop, and mounting wheels can be tricky. The small plastic gearboxes work well if you can get matching wheels but are hard themselves to mount. Access to a 3d printer can be quite useful for creating missing physical components, though a block of wood and a router can make a nice chassis, too. You can also buy ready made Arduino robot base kits with chassis, gearboxes, and wheels.
For audio output, you can accomplish something with an NPN transistor and base resistor driving a speaker, but best results would be with a small class D driver board as are now widely sold for a few dollars. They tend to be stereo, but you can feed both sides with the same signal. If you only wanted "robot beeps" you can drive a piezo disk directly from a processor pin with reasonable loudness at higher frequencies, but the low frequency response is quite poor.