Dealing with millions of data records in MySQL and PHP/Laravel
I would recommend to take a look at the https://www.elastic.co/products/elasticsearch and save your activity records to its storage when you do save to the main database. Then you can easily search any field. Elasticsearch can store a schema free JSON documents, if you prefer more SQL way, there is another search engine - http://sphinxsearch.com/.
There is no problem inserting a zillion rows into a table. Performance problems come when you try to do non-trivial SELECTs on the table. You mentioned "search"; you will have to limit what the 'users' can search for. But at least make a stab at what they might want to search for.
You mentioned "searching for an object", but I don't see a column called object. How many rows might there be for a given object? Do you need all the rows? Or selected ones? (An INDEX on object is likely to make the query efficient, regardless of table size.)
Third-party software sometimes gets in the way of dealing with really large tables. Beware.