A new router may not help. The right answer for you is going to depend a lot on the layout and construction of your home, how much work you want to put in, and your budget. It’s also important to understand how wifi works. Wifi is half-duplex, meaning that it can only send or receive (not both) to one device at a time; wired Ethernet, however, is full-duplex. The importance of that will become evident later. The short answer is that there are a lot of ways to improve wifi coverage/speed/reliability, but each has its pros and cons.
Installing additional access points with wired backhauls - and powered via Power-over-Ethernet (PoE) instead of 120VAC - to your main router is ALWAYS going to be the best solution in terms of reliability and performance. If you have a single story home and/or sufficient attic access, you can get this done in half a Saturday, and Cat6 is cheap. However, this obviously requires the most work, and since the APs will likely be on different channels you will not have completely seamless handoffs from one AP to the next as you walk through your house.
Repeaters or range extenders are like access points, just with wireless backhauls. These can work in the right situations, e.g. you have a dead spot in one part of the house, or like HH’s porch. You wouldn’t want to push a ton of traffic through it, though - the half-duplex connection back to the router can become overburdened. And you certainly wouldn’t want to connect an extender to another extender.
Mesh systems are like range extenders on steroids, and are priced accordingly. The nodes are designed to be small and relatively discreet so they can be placed almost anywhere, and these can work well in the right situations and if the base station and nodes are placed correctly. For instance, my parents’ two-story house is an odd configuration and the Internet connection is on the far end of the house. A mesh system - with one node downstairs and one up, each connected to the base - worked great for them. I’ve seen some mesh systems are now including a third wifi radio dedicated to backhaul communication, which is a neat way to mitigate (but not solve) the half-duplex problem. Mesh systems also typically operate on a single wifi channel, so your phone/tablet/computer will maintain its connection better as you move about the house.
Another easy solution is to use a powerline Ethernet adapter to get Ethernet where you need it, then connect an AP. Powerline adapters transmit Ethernet over your home’s electrical wiring. If you want the benefits of a wired AP without sweating your ass off in the attic, it’s not a terrible solution. You’ll take a big hit on throughput - powerline Ethernet can only do 50-100mbps, vs. 100-250mbps for a mesh/extender and 1000mbps for wired Ethernet) - but it would be fast enough for Netflix and such.