I would suggest Ancestry. They have an entirely free option that allows you to create your own tree and add information to it, but it does not allow you to search any of their information (apart from that which is free anyway)
You can sign up for free here:
https://support.ancestry.co.uk/s/article/Free-Registered-Guest-Accounts
The records that can be searched for free are here:
https://www.ancestry.co.uk/search/categories/freeindexacom#included-data-collections
In order to let other people see your tree on Ancestry then they would also need to open a free account and then you can invite them as a guest to view your tree.
If your tree is quite large then you can upload it to your free Ancestry account as a GEDCOM file (Family Tree Maker will likely have that capability).
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"I believe his son (my cousin) has also done some of the tree so would say Ancestry match us up?"
Is that on Ancestry or some other place? If he has a tree on Ancestry then he can invite you to view his tree.
You can then do the following with a paid account, but I don't know if you can do this with a free account. It's the nearest you'll get to merging two trees, although it's quite manual.
If you go to a record that is in both your trees you can then save all the information from his tree directly to yours. Not only for that person but also their spouse and children as well.
So, for example, if on both trees you both have Joe Smith. But on your tree you only have Joe. In contrast, your cousin's tree also has Joe's wife and children as well.
If you were to save Joe's details to your tree then you will have the option to also save the details of his wife and children as well. They will then be added in the correct place on your tree.
It's very manual, but at least you don't have to add everybody individually.