Vostro 3560 max ram

The maximum per Dell specs and Crucial is 8GB (2 x 4GB). Now having said that even though the official specs call for a maximum of 8GB, this forum thread has users confirming they have installed 16GB (2 x 8GB) successfully.

There are potentially multiple causes for why your laptop now runs more slowly.

RAM

You asked about RAM, and 4 GB RAM is inadequate. Crucial and Kingston both guarantee that your laptop can use maximum 8 GB of their RAM. As @JOcean found, some users report that they use 16 GB, so that might work. Given the other specs of the laptop, I think 8 GB is a worthwhile upgrade, while you may not need 16 GB. Here is RAM that Crucial guarantees compatible with your Vostro.

You may try other RAM brands, as long as they match those specifications. I recently bought a relatively unknown brand, Kuesuny, of DDR3L RAM (the same as you need) through Amazon for much less than the price of a big brand name. I'm happy with it. I ran memory tests for a couple of days to thoroughly test the RAM when I first installed it. It performs well and saved me money.

SSD

You didn't say whether it has a hard drive or an SSD. If it's a hard drive, replacing it with an SSD will make it much faster and more responsive, a noticeable difference and the most bang-for-your-buck that you can get for any upgrade. That will likely make a bigger difference to the performance than the RAM upgrade.

You may use any 2.5" SATA SSD up 4 TB if you have a UEFI system, or 2 TB with legacy BIOS, either 7mm high (you may need a plastic spacer/adapter, which Crucial includes) or 9.5mm high. Here are Crucial SSDs that would work in your Vostro (click on the Internal SSD tab). 

You may not need a spacer/adapter for the SSD. I recently installed 7mm high SSDs into two old Dell XPS laptops. The bay originally held 9.5mm HDDs. While I can add the 2.5mm spacer/adapter that Crucial included with its SSDs, I found that it's not required and wouldn't add any benefit. The original drive bracket holds the new SSD solidly and securely once all four screws are tightened.

Other potential causes of slowdowns on your laptop:

Thermal compound

Your laptop is ten years old. The thermal compound has dried out and cannot adequately transfer heat away from the processor, and there may be a lot of dust inside the laptop and maybe inside the fan and heat fins. So, your laptop may be unable to adequately cool itself. That would cause the processor to throttle (slow down).

Clean out dust

You might consider an internal cleaning. Open it up to clean out dust. Disassemble the fan to clean out dust and fluff. I found a fair bit in two laptops I worked on recently. Remove the heat sink, remove the old thermal compound and apply new thermal compound before reinstalling the heat sink.

CMOS battery

While you're in there, replace the CMOS battery (coin-cell battery). If it isn't dead, it soon will be. BIOS settings and date/time can't be saved without a functioning CMOS battery.

Safety and procedures

Follow the disassembly and reassembly steps in the Vostro's Owner's Manual.
Look on YouTube for videos showing disassembly of your model. But, be cautious with them, as some use steps or methods that are not appropriate. Use the videos to add to your understanding of what to do, but follow the steps in Dell's manual.

Whenever touching components or working inside a computer, wear a grounded wrist strap, also called anti-static wrist strap, ESD wrist strap, or ground bracelet. It's a cheap and sensible precaution. Rest the laptop on an anti-static mat or at least a reasonable alternative such as corrugated cardboard. (Repeatedly touching a metal case part is not sufficient mitigation with modern components.)

Windows 10

Also, what OS are you running? If you have 8.1, 8 or 7, you should back up your data to an external drive or flash drive, and then perform the free upgrade to Windows 10.  After upgrading, check to see if Windows 10 is activated in Settings | Activation. If it is, then you can perform a clean install of Windows 10 and it should activate automatically. The clean install will result in a better system than just staying with the in-place upgrade. You could do the free upgrade on your current drive, after which you can install a new SSD and perform the clean install on it.

Here are my detailed steps to perform a clean install of Windows 10. And, here's another description of the install process, from a Microsoft MVP.

I realize that seems like a lot of work. You must judge if it's worthwhile to you. You could buy a newer, faster system. However, if you want to keep the system going, this work will make it faster and more reliable.

Thank you for your very full explanation and suggests Filbert.

I'm not 100% sure if my Vostro has a Hard Drive or a SSD but I have just typed "hard drive" into the windows search bar (bottom left of screen) and it gave me some info I didn't understand: data about a Disk O basic 465.76GB online and OS (C:) of which 38% is free.

No info came up when I searched "SSD". so I'm guessing my Vostro has a "hard drive".

I clicked your link and clicked internal SSD and it took me to directly to the 4TB option.  Would I need 4TB in a laptop I only use for data, word and emails? or would I get away with 1TB?

I'm pretty sure it wasn't normal for laptops to have even 1TB of memory when I bought my Vostro.

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@JW1963 wrote:

I'm not 100% sure if my Vostro has a Hard Drive or a SSD

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It's easy to find out. Open Windows app Defragment and Optimize Drives. If you press the Windows key and type defrag then the app will likely appear in the menu so you can select it. It will tell you if you have a hard drive or a solid state drive.

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@JW1963 wrote:

I clicked your link and clicked internal SSD and it took me to directly to the 4TB option.  Would I need 4TB in a laptop I only use for data, word and emails? or would I get away with 1TB?

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Buy what you need. You can go all the way to 4 TB if you have a UEFI system, or 2 TB with legacy BIOS, but probably you need only 1 TB. You might get away with only 500 GB, if you clean out a lot of unnecessary stuff that currently eats up space in your drive. If you do need all of that then buy a 1 TB drive.

Thank you

I have done that and it says "defrag your hard drive" so I guess that answers my question. The label on the drive that is in the laptop atm says SATA 6.0b/s capacity 500gb RPM 7200 so I guess that also confirms the size as half a TB.

I have ordered a 1TB SSD; will it be "plug and play" or will I have to do anything else?

And will all my existing document files, photos, music and other programs be unaffected by the change from hard drive to ssd?

Sorry for the dumb questions but I'm not really sure which component of my laptop does what.

Regards

JW1963