During your very first week, you have a very limited selection of actions which gradually expands over time. When opening up your shop, for the first day or three, you can only sell items you have on display; there's only one major dungeon that you can access (other than the tutorialized one), and shop selection is limited by the poor allowance you have. How are we going to get around this?
1. Go to dungeons at least once a day.
Because you can only exit the dungeon after every boss room (every fifth floor), you need to rationalize whether what you have will be worth the trip. Lots of chestnuts and slime? They aren't going to be very sellable, so throw away the lower-quality ones, keeping only the +3s and +4s. Water Crystals? HOLD ONTO THEM. Basically, hold onto the drops that you rarely get, such as Water Crystals, Honey, Bat Wings (more on that later). Also, keep the higher-tier items that you find. The best you can encounter at this point has typically been the Steel Edge, but this is rare.
2. Sell for close to 106%.
Right now, you have a very low merchant level, and you can only buy items. To level up quickly, you'll want to sell at around 106%, as characters will purchase at that price most of the time. For every purchase you get on the first price, you increase your Just combo. At the first customer, it's only worth 2 exp, but each subsequent first-purchase doubles this, to a limit of 128! Also, 106% is close to the Pin bonus for some characters. Getting a Pin bonus (nailing the price they want) will earn you a bonus 30 exp, and getting a Near bonus (getting close to the price they want) will earn 15 exp. You can see that both of these concepts lead to one conclusion: even if the items are only worth 100pix each, it would be a good idea to sell for close to the value.
3. BUY CANDY.
In the second week, a new mechanic will be introduced, which are Booms and Busts. The very first one that will be introduced uses Candy, which can be purchased from the market. Make sure to buy a good supply of Candy for this event; Sales are part of the Week 2 ideas.

That's it for week 1!

By now, you have probably beaten the first dungeon a couple times, and gotten the expensive vase. If you've sold it, then congratulations! You are likely at least halfway through this week's payment! Additionally, if you've befriended Charme enough, and visited the townfolk in the bar, square and market enough, she'll invite you to a new dungeon. Here, you can get some new equipment levels, as well as a new character for beating it.
Quick tips for character choice: if you are used to simply hacking-and-slashing your way through a dungeon by now, stick with Louie. However, if you find yourself getting impatient at his slow movement speed, try out Charme. She can dash through dungeons quickly, but is more of a glass cannon. Additionally, she doesn't have the same shield to protect from projectiles. However, she does have a special move that makes you invulnerable for a short period of time. Either way, it's up to you.
Tips for increasing your cash yield:
Stick with the lower prices for now.
As soon as you start rising prices, you lose out on those valuable combos. Besides, in this first run through the game, you are not likely to make it completely clean, and will need to start again. More of this will be explained next segment, but for now, keep focusing on raising your level.
Buy low, sell high!
Remember how in week 1, I suggested you hoard Candies? Now it's time to put them to use! Partway through the week, there'll be an event where text runs across the top of the screen saying Candy is high in demand. PUT OUT ALL YOUR CANDY ON THE TABLES. You can sell for more than 200% the value, and still keep a combo going. True, it isn't anywhere near the Pin value for most characters, but you'll still keep the combo, and rack up exponential experience. There is a dark side to these booms, though: Other shops will also be selling for higher prices. So don't stock up on items during booms.
On the flip side of the coin, there are depressions: times when items will be decreased in value. These are opportune times to stock up on high-value items or other highly-demanded merchandice. In another note, TAKE YOUR DEPRESSED ITEMS OFF THE SHELVES. You will not be able to sell them for anywhere NEAR 100% during these times!
Do not visit the Town Square if your pix level has gone past 100,000
Doing so will trigger the worst NPC that can enter your store: Euria. She doesn't buy anything. She doesn't sell items that are worth the 500% price she asks for. The only reason you would want to trigger her events is if you're looking for the few items that can't be found in ruins or the shops. Again, DO NOT VISIT THE SQUARE IF YOU HAVE >100,00 PIX. Haven't visited the Square to trigger Tielle's event and the third dungeon opening? Tough luck. Go buy some stuff until you have <100,000, and come back later.
That's all for Part 2! Next up: the first major difficulty spike of the game.