Yes, start with a strong foundation by learning hiragana and katakana early, and practice reading and writing them diligently.
Don't try to rush it too much, slowly familiarize yourself with them by working on one or two new sets* a day. Practice reading and writing them just before bed, and again in the morning, to help encode them in your long-term memory. As you add each new set, be sure to refresh what you've already learned and write out the previous sets. So at first you're just writing the five vowels (say 10-20 times), but at the end of a couple weeks, you'll be practically writing out the entire phonetic alphabet. Get away from romaji as fast as possible.
You can try to study hiragana and katakana together, but I'd recommend focusing on hiragana first and then tackle katakana, They represent the exact same sounds but are used for different things,. It's like if English were written in all lower case letters, but words that are EMPHASIZED, sound effects like BOOM and foreign loan words like SUSHI were written in all capitol letters.
Equally important to the rote memorization of just learning all these new characters is to learn how they are used; Genki will introduce new vocabulary that uses the characters you've learned so you can see them in a real world. This will help your brain build connections between the characters, the sounds they make, and the words they appear in.
*by set, I am referring to a group of the five vowel sounds starting with a particular consonant, ie
あ い う え お - a i u e o (no consonant)
か き く け こ - ka ki ku ke ko
ま み む め も - ma mi mu me mo
and so on. There are a couple special characters you will encounter, and you should focus on those each in turn when you are ready.
