If Est. sales for Oct, Nov, Dec and Jan are as follows:
$18,441.00 $22,890.00 $48,505.00 $19,393.00
And the COGS is on average about 60% of sales
What all else do you need to know to create a purchasing budget for those month?
The Beginning Inventory Estimation of those months?
Also, is there a set formula for deciding how much you actually need to buy to make those sales, since you will want to buy more than what you actually sell?

Yes, there's an inventory adjustment thing to be done. But first you need to actually get the 60% of the sales so that you'll know what the COGS will be for each month.

COGS
+ desired ending inventory
- beginning inventory
= Purchases need to be made

The COGS is the cost of what you think you'll sell. But, as you said, you'll want to have more than you'll sell, so you add on the desire ending inventory. But, if you already have some (beginning inventory) you don't need to buy those, so that subtracts back off.

The problem should contain the first beginning inventory, and probably have some way for you to calculate the desire ending inventory. Then just remember that ending inventory for one month is beginning inventory for the next.

That helps a lot!
Actually, this is a real scenario in my life, not a school thing.
I have my current inventory valuation, and my numbers from LY.
Last year was our first year in our current location, so I only have the 1 year to base our previous inventory valuations on (on 1st day of month).
Should I use these to estimate the beginning inventories for the future months? (+ 10% since our sales are expected LY + 1-%)?
I'm having trouble wrapping my brain around creating this budget. But if you know what I am overlooking, that might really help!

Um... creating budgets can be difficult, especially from the outside when I don't know your business. It's also a managerial accounting area, which is not my specialty area.

You need to decide how much you want as a desired ending inventory. The beginning inventory numbers will always just carry forward each month. The only "formula" I'm aware of for finding an appropriate desired inventory is pretty complicated, involves statistics, and something I only learned in a class, hated then, and don't remember now. But that formula is meant to try to create the "perfect" inventory amount based on equalizing the costs of carrying inventory versus running out of inventory.

Past years are always a good place to start. If you thought your inventory was appropriate, you can certainly use that as a starting point (including your 10% increase if you think that's realistic). If you don't think it was appropriate, adjust it. You may even ask others in your business what they do, because it will certainly depend on the type of business you have. (Perhaps search for some online trade magazines in your business area, which might also have forums, etc.)

Do take into consideration the cost of carrying inventory. i.e. product going bad if it's perishable, or even affected by aging, product becoming obsolete, storage space, property taxes, insurance, costs of tracking it, etc. Versus how it would hurt you if you ran short.

Again, managerial accounting isn't my specialty area. But that's some stuff to think about.